COMPREHENSIVE

Section 1. ТНЕ PRACTICE AND THEORY OF КINETICS (3 volumes)

Section 2. HOMOGENEOUS DECOMPOSITION AND ISOMERISATION REACTIONS (2 volumes)

Section 3. INORGANIC REACTIONS (2 volumes)

Section 4. ORGANIC REACTIONS (6 volumes)

Section 5. POLYMERISATION REACTIONS (3 volumes)

Section 6. OXIDATION AND COMBUSTION REACTIONS (2 volumes)

Section 7. SELECTED ELEMENTARY REACTIONS (1 volume)

Section 8. HETEROGENEOUS REACTIONS (4 volumes)

Section 9. КINETICS AND CHEMICAL TECHNOLOGY (1 volume)

Section 10. MODERN METHODS, THEORY, AND DATA

CHEMICAL KINETICS

EDITED ВУ

R.G. COMPTON

М.А., D.Phil. (Охоп.) University Lecturer in Physical Chemistry

and Fellow, St. John's College, Oxford

VOLUME 32

КINETIC MODELS OF CATALYTIC RЕАСТЮNS

ELSEVIER AMSTERDAM-OXFORD-NEW YORK-TOKYO

1991

ELSEVIER SCIENCE PUBLISHERS B.V. Sara Burgerhartstraat 25

Р.О. Вах 211, 1000 АЕ Amsterdam, The Netherlands

Distributors for the United States and Canada

ELSEVIER SCIENCE PlJBLISНING СОМР ANY INC. 655 А venue of the Americas

New York, NY 10010 U.S.A.

ISBN 0-444-41631-5 (Series) ISBN 0-444-88802-0 (v 01. 32)

with 81 illustrations and 10 tables

© Elsevier Science Publishers B.V., 1991

АН rights reserved. No part of this publication тау Ье reproduced, stored in а retrieval system or transmitted in any form or Ьу any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission ofthe publisher, Elsevier Science Publishers B.V., Physica! Sciences and Engineering Division, Р.О. Бох 330, 1000 АН Amsterdam, The Nether!ands.

Specia! regu!ations for readers in the USA - This publication has been registered with the Copyright C!earance Center Inc. (ССС), Ва!ет, Massachusetts. Information сап Ье obtained пот the ССС about conditions under which photocopies of parts of this publication тау Ье made in the USA. АН other copyright questions, inc!uding photocopying outside of the USA, shou!d Ье referred to the publishers.

Printed in the Nether!ands

COMPREHENSIVE CHEMICAL КINETICS

ADVISORY BOARD

Professor С.Н. BAMFORD

Professor S. W. BENSON

Professor LORD DAINTON

Professor G. GEE

Professor G.8. HAMMOND

Professor W. JOST

Professor к.J. LAIDLER

Professor SIR HARRY MELVILLE

Professor S. OKAMURA

Professor N.N. SEMENOV

Professor Z.G. SZABO

Professor О. WICHTERLE

Contributors to Volume 32

АН chapters in this volume have Ьееп written Ьу

V.I. BYKOV Computer Centre, Siberian Branch of the U.8.8.R. Academy of Sciences, Akademgorodok, 660036 Krasnoyarsk 36, U.S.S.R.

V.I. ELOKHIN Institute of Catalysis, Siberian Branch of the U.S.S.R. Academy of Sciences, Prospekt Akademika Lavrentieva 5, 630090 Novosibirsk 90, U.S.S.R.

A.N. GORBAN' Computer Centre Siberian Branch of the U.S.S.R. Academy of Sciences, Akademgorodok, 660036 Krasnoyarsk 36, U.S.S.R.

G.8. У ABLONSKII Touvinian Complex Department, Siberian Branch of the U.S.S.R. Academy of Sciences, Prospekt Lenina 30, 667000 Kysyl, U.S.S.R.

Preface

This volume is concerned with providing an up-to-date and rigorous account of the dynamics of catalytic reactions. The authors' view of the detailed objectives and scope of the book are summarised in their preface which follows.

Oxford November 1990

R.G. Compton

Authors' Preface

"Yet who would Ьауе thought the old man to Ьауе had во тисЬ blood in him?" This title, given Ьу Prof. Rutherford Aris and his collaborator W.W. Farr to their recent paper [Chem. Eng. Sci., 41 (1986) 1385], is а phrase used Ьу Lady Macbeth (Macbeth, V, 1, 42-44). Fierce, isn't it? Apparently, they mean it to imply that traditional theoretical problems in the dynamics of chemical reactions, in particular the known problem of the dynamics of the continuous stirred tank reactor (CSTR), are far from being exhausted. Novel mathematical approaches provide new results oriented to physico-chemical comprehension. This current trend is confirmed Ьу the present volume.

This book Ьав been written Ьу mathematicians and chemists, the col­laborators of the Institutes of the Siberian Branch of the U.S.S.R. Academy of Sciences [ТЬе Institute of Catalysis (N ovosibirsk), the Computing Centre (Krasnoyarsk) and the Тиуа Complex Department (Kyzyl)]. It presents the results of 15 years activity of this Siberian team ав reported in two earlier monographs (Kinetic Madels а{ Catalytic Reactians, Nauka, Novosibirsk, 1983 and Kinetics (ar Madel Reactians а{ Heterageneaus Catalysis, Nauka, Novosibirsk, 1984, both published in Russian). Unfortunately, these results are hardly known to English-speaking readers.

Compared with the аЬоуе monographs, the book ЬаБ been revised and completed. It accounts for the rapid development of events in the region of mathematical chemistry, which is at а crossroads where chemistry, physics, mathematics and chemical technology meet. Besides establishing а general theory permitting ив to investigate the dependence ofkinetic characteristics for complex reactions оп the structure of the detailed mechanism, the book provides а comprehensive analysis of воте concrete typical mechanisms for catalytic reactions, in particular for the oxidation of carbon monoxide оп platinum metals. This reaction is а long-standing traditional object of chemi­cal catalytic investigations. In fact, the book presents "three kinetics"; (а) detailed, oriented to the elucidation of а detailed reaction mechanism ас­cording to its kinetic laws; (Ь) applied, with the aim of obtaining kinetic relationships for the further design of chemical reactors; and (с) mathemati­cal kinetics whose purpose is the analysis of mathematical models for heterogeneous catalytic reactions taking place under steady- or unsteady­state conditions.

What unites the Siberian team? Primarily it is а common interest in the complex dynamics of catalytic reactions. ТЬе boundaries of this field are

extended and dynamic approaches are applied to а variety of chemical systems. The newest and, as it would seem, absolutely abstract mathematical results make it possible not only to answer questions that are traditionally interesting for chemists and physicists, but also to extend their list.

The authors are extremely grateful to their colleagues, theoretical and experimental workers, Drs. V.V. Barelko, V.M. Cheresiz, V.A. Evstigneev, A.I. Ivanova, V.I. Marshneva, V.I. Savchenko and A.I. Vol'pert for the fruitful collaboration and discussions without which this book would not have Ьееп written. The authors also tЪапk I.I. Sochelnikova for the transla­tion of this composite mathematical-chemical manuscript.

Grigorii У ablonskii Valerii Bykov Aleksandr Gorban' Vladimir Elokhin

Contents

РгеСасе ..... Authors' ргеСасе

Introduction References.

Chapter 1

Minimum minimorum . . . . . . . 1. Introduction. . . . . . . . . 2. Chemica! kinetics and !inear a!gebra. 3. Unsteady- and steady-state kinetic mode!s 4. Steady-state reaction theory. . . . . . . 5. E!ements of qua!itative theory of differentia! equations 6. Re!axation in cata!ytic reactions

References. . . . . . . . . . . . . . .

Chapter 2

The deve!opment оС basic concepts оС chemica! kinetics in heterogeneous cata!ysis ............................... . 1. Steps in the deve!opment of genera! chemica! kinetics. . . 2. The deve!opment of the kinetics of heterogeneous cata!ysis

2.1 Idea! adsorbed !ayer mode! . . . . . . . . . . . . . 2.2 Rea! adsorbed !ayer mode!s . . . . . . . . . . . . . 2.3 Mode!s accounting for phase and structura! transformations

of cata!ysts . . . . . . . . . 2.3.1 Phenomeno!ogica! mode! 2.3.2 Lattice gas mode!. . . . 2.3.3 Topochemica! mode!s. .

2.4 Mode!s accounting for diffusiona! mass transfer 2.5 Heterogeneous-homogeneous cata!ytic reaction mode!s . . . . . . 2.6 Phenomeno!ogica! mode! of branched-chain reactions оп а cata!yst

surface 3. Conc!usion . . . .

References.

Chapter 3

Formalism оС chemica! kinetics. 1. Main concepts of chemica! kinetics

1.1 Linear !aws of conservation. . 1.2 Stoichiometry of complex reactions 1.3 Graphica! representations of reaction mechanisms. 1.4 Chemica! kinetics equations. 1.5 Reaction po!yhedron . . . . . . . . . . . . . .

lX

Х

5

7 7

11 16 20 29 39 45

47 47 59 60 61

64 64 67 71 75 78

79 79 80

85 85 85 86 88 95 98

1.6 Reaction rate . . . . . 105 1.7 Concentration equations 106 1.8 Non-idea! systems . . 109

2. Princip!e of detai!ed equi!ibrium and its corollaries 112 2.1 Princip!e of detai!ed equi!ibrium. . . . . . . 112 2.2 ТЬе uniqueness and stabi!ity of equi!ibrium in c!osed systems 120 2.3 Thermodynamic !imitations оп non-steady-state kinetk behaviour 130 2.4 Limitations оп non-steady-state kinetic behaviour imposed Ьу the reac-

tion mechanism . . . . . . . . _ . . . . 135 3. Forma!ism of chemica! kinetics for ореп systems 140

3.1 Кinetic equations for ореп syste~s. 140 3.2 "Weak!y ореп" systems. . . . . . 147 3.3 Stabi!ization at high flow ve!ocities 150

4. Quasi-stationarity . . . . . . . . . . 152 5. Uniqueness, mu!tip!icity and stabi!ity of steady states . 164

5.1 Linear mechanisms. . . . . . . . . . . . . . 165 5.2 Mechanisms without intermediate interactions . . 169 5.3 Quasi-thermodynamic Horn and Jackson systems . 174 5.4 Criterion for uniqueness and mu!tiplicity associated with the

mechanism structure 177 5.5 Some conc!usions. 182

References. 183

Chapter 4

Graphs in chemical kinetics . . . . . . . . 185 1. Genera! description and main concepts. 185

1.1 Simp!e ехатр!е . . . . . . . . . 185 1.2 Two formalisms. Forma!ism of enzyme kinetics and of steady-state

reaction theory. . . . . . . . . . . 190 1.3 Non-!inear mechanisms оп graphs. . 198

2. Graphs for steady-state kinetic equations . 199 2.1 Substantiation of the "Mason rule" . 199 2.2 Genera! form of steady-state kinetic equation [or сотр!ех cata!ytic

reactions with mu!ti-route !inear mechanisms. . . . . . . . . .. 202 2.3 Ana!ysis of properties for the genera! steady-state kinetic equation of

сотр!ех catalytic reactions. . . . . . . . . . . . . . . 205 2.4 How to find the kinetic equation far reverse reactians. . . 214 2.5 Matching of reactions and the representation of the kinetic

equation in the Horiuti-Bareskov farm. . . . . . . . . . . . . . . . 217 2.6 Observed kinetic regu!arities and characteristics of detailed

mechanisms . . . . . . . . . . 2.6.1 Observed reaction order. . . . . . . . . . . . 2.6.2 Observed activation energy. . . . . . . . . .

3. Graphs for the analysis of the number af independent parameters. 3.1 Simp!e ехатр!ев . . . . . . . . . . . . . . . . . . . . . 3.2 Reasons for dependence and the impossibi!ity of determining

219 219 225 229 229

parameters. . . . . . . . . . . . . . . . . . . . . . 231 3.3 Indeterminacy of parameters and graph structure. . . . . _ 234 3.4 ТЬе number of determinable parameters and graph со!оиг . 238 3.5 Brutto-reaction, detai!ed mechanism and the number af parameters

under determination . . . . . . . . . . . . . . . 241 3.5.1 Brutto-equation and the number of steps. . . 244 3.5.2 Graph co!ours and kinetic equation structure _ 245

4. Graphs to analyze relaxations. General form of characteristic polynomial .

5. Conclusion . . . References.

Chapter 5

Simplest non-linear mechanisms of cata!ytic reactions producing critica!

251 256 257

phenomena. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 259 1. Critica! phenomena in heterogeneous catalytic reactions

(experiment, models) . . . . . ., . . . . . . . . . . 259 2. ТЬе "parallel" adsorption mechanism . . . . . . . . . 271 3. Steady-state characteristics of the simplest mechanism permitting

multiplicity of catalyst steady states. . . . . . . . . . . . . . 277 4. Relaxation characteristics of the "parallel" adsorption mechanism 285 5. Analysis of "consecutive" adsorption mechanisms. . . . . . . . 295 6. Models of kinetic self-oscillations in heterogeneous catalytic reactions 298

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

Chapter 6

Studies of kinetic mode!s for oxidation reactions over meta!s (exemplified Ьу СО oxidation). . . . . . . . . . . 311 1. Mechanisms and models . . . . 311 2. Modelling of kinetic dependences 322 3. Dynamic studies of СО oxidation 338 4. "General" kinetic model and prediction of critical effects 353

References. . . . . . . . . . . . . . . . . . . . . . , . . . 356

Chapter 7

Critical retardation effects and slow relaxations 1. ТЬе ргоЫеm of slow relaxations. . . . . . 2. ТЬе limit behaviour of dynamic systems . . 3. Relaxation times. Determination of slow relaxations. 4. Bifurcations (explosions) of limit sets . . . . . . . 5. Dynamic factors for slow relaxations. . . . . . . . 6. Taking into account Бmаll perturbations and errors of models 7. Conclusion . . . . .

References.

Chapter 8

Conclusion. . . . . . . . . . . . . . 1. Forecast for tomorrow . . . . . 2. Afterthoughts to the conclusion .

References.

Index ...

361 361 364 366 370 372 376 380 380

383 385 386 387

389

Introduction

This book embraces problems having both physicochemical and mathema­tical sense. А widespread concept of "mathematical physics equations" includes pri­

marily linear and quasi-linear partial differential equations. But wh&t are "mathematical chemistry equations" and "mathematical chemistry" in general?

The concept of "mathematical chemistry" had been already used Ьу M.V. Lomonosov [1] and later оп in the 19th century Ьу Du Bois-Reymond, but for а long time it Ьесате inapplicable, apparently due to the lack of а distinct field for its application. АБ а rule, it was, and has remained, preferable to speak about the application of mathematical methods in chemistry rather than about "mathematical chemistry". То our mind, it is now quite correct to treat mathematical chemistry аэ а specific field of investigation. Its equations are primarily those of chemical kinetics, i.e. ordinary differential equations with а specific polynomial content. We treat these equations relative to heterogeneous catalytic systems.

We will try to define the way in which the structure of а complicated chemical mechanism and its respective kinetic model are associated with the peculiarities of steady and non-steady kinetic relationships, i.e. how the elementary reactions with simple kinetic dependences lead to complicated kinetic behaviour.

Chapter 1 presents аН the necessary information concerning linear alge­bra and the qualitative theory of differential equations in terms of which we construct and analyze kinetic models of heterogeneous catalytic reactions.

Chapter 2 describes the evolution in fundamental concepts of chemical kinetics (in particular, that of heterogeneous catalysis) and the "prehis­tory"of the problem, i.e. the period before the construction of the formal kinetics apparatus. Data are presented concerning the ideal adsorbed layer model and the Horiuti-Temkin theory of steady-state reactions. In what follows (Chapter 3), an apparatus for the modern formal kinetics is re­presented. This is based оп the qualitative theory of differential equations, linear algebra and graphs theory. Closed and open systems are discussed separately (аБ а rule, only for isothermal саБеБ). We will draw the reader's attention to the two results of considerable importance.

(1) Many kinetic problems of complex reactions in ciosed systems сап Ье treated from the geometrical viewpoint. In due course it Ьесате possible for

References рр. 5-6

2

analytical mechanics; now it сап also Ье realized for chemical kinetics. Geometrical representation not only simplifies the interpretation, but also offers new possibilities, e.g. for planning experiments.

(2) Sets of quasi-steady-state equations сап have several solutions, which correspond to several steady-state rate values of complex reaction in open systems (multiplicity of steady states). It has Ьееn shown that the necessary condition here is the presence о! аn interaction step between various inter­mediates in the complex reaction mechanism. Let ив discuss this result in more detail.

Since the late 1930в (studies Ьу D.A. Frank-Kamenetskii, Уа.В. Zeldovich and N.N. Semenov), it has Ьееп known that there exist multiple steady states of chemical systems due to the simultaneously occurring chemical, thermal, and diffusional processes [2]. It is во evident that it сап Ье shown very simply. Thus, in а non-isothermal continuously stirred tank reactor (CSTR) the temperature dependence of the rates for heat release and heat removal are exponential and linear, respectively. These dependences сап Ье balanced at more than опе point, and it is this fact that accounts for several steady states. In non-isothermal cases one сап also observe self-oscillations of the reaction rate (including first-order reactions). Generally speaking, in поп­isothermal савев the critical effects of various types, i.e. multiple steady states and self-oscillations, have Ьееп thoroughly investigated both experi­mentally and theoretically as early as in the 1930s-1940s. А new fact to Ье accounted for was that similar effects had been found in а purely kinetic region, i.e. in isothermal conditions without а distorting effect of heat and тавв transfer. At the present time а great number of experimental data have Ьееп accumulated for isothermal critical effects. Thus ав early as in the 1950в, Boreskov et al. [3] established а critical effect for the kinetics of hydrogen oxidation оп nickel catalysts. Under isothermal conditions with certain gas-phase compositions, а "downward jump" in the reaction rate is observed. The same gas composition accounts for drastically different reac­tion rates. It is just а case of multiple steady states. Also changeable (from first to zero) is the reaction order. Similar phenomena had also Ьееп found for the kinetics of СО oxidation and hydrogenation. Apparently, Davis [4] was the first to observe these phenomena (in the 1930в), but his experimental data were met with distrust. Finally, and more recently, after the develop­ment of а special electrothermographical technique Barelko and Merzhanov revealed the existence of critical effects being purely kinetic in nature [5]. They established that тапу complete catalytic oxidation reactions сап lead to such kinetic "revolutions". For ammonia oxidation оп platinum they observed hysteresis: the transition from "upper" to "lower" steady state and back takes place at different values of the reaction parameters. The kinetic dependences obtained Ьу increasing and decreasing the parameters are different.

In the 1950в-1960в interesting experimental data were obtained for воте homogeneous catalytic reactions. Late in the 1950в, Belousov established

3

isothermal self-oscillations for the catalytic oxidation of citric acid Ьу bro­mate (catalyzed Ьу cerium ions). For воте time, Belousov's paper appeared in а small-circulation publication and was little read Ьу chemists. It was thus largely ignored. Оп S.E. Shnol's initiative, Zhabotinskii et al. continued the investigation of this reaction. They showed experimentally that self-oscilla­tions do exist due to the complex reaction chemistry and they constructed а qualitative mathematical model based оп the "law of тавв action" [6]. Кееп interest was shown in the Belousov-Zhabotinskii reaction in par­ticular and in chemical self-oscillations in general. Several hundred papers were devoted to the Belousov-Zhabotinskii reaction.

According to Zhabotinskii, а simplified reaction scheme in the system consisting of malonic acid (МА) + bromate and cerium ions is of the form

BrO­СеЗ+нf-+ Се4 +

Се4 + ~ СеЗ +

(1)

(2)

Reaction (1) yields products of ВrОз reduction (Br2 , HOBr) which bro­minate МА to form its bromine derivatives (ВМА). Reaction (2) produces ап inhibitor for reaction (1) acting ав а feedback. If the system contains much Се4+, the amount ofBr- is also high and reaction (1) is hindered. Finally, the amount of Се4+ [аllв to its lower critical value and the concentration of Br­also decreases. Reaction (1) then proceeds at а high rate and everything begins again.

Self-оsсШаtiопs have also Ьееп revealed for heterogeneous catalytic reac­tions. Hugo and Jakubith [7] and Wicke and co-workers [8] found self-oscilla­tions for СО oxidation оп platinum. In the period 1973-1975, M.G. Slinko and co-workers studied self-oscillations in hydrogen oxidation оп nickel [9,10].

Belousov-Zhabotinskii experimental data Ьесате опе of the starting points for Prigogine and his school in studying the complicated dynamic behaviour of "far from equilibrium" chemical systems. These investigations were reported in а series of monographs [11,12] and Prigogine has recently Ьееп awarded а Nobel prize. То interpret isothermal critical effects quantitatively, Prigogine used

mechanisms involving autocatalytic steps, i.e. а 'Ъrussеlаtоr" and ап "or­egonator". These патев have Ьееп derived from "Brussels" + "oscillator" and "Oregon" + "oscillator", since various research groups worked in Brussels and Oregon. It must Ье emphasized that, ав long ago ав the 1940в, Zeldovich reported that ап autocatalytic reaction under isothermal соп­ditions сап also lead to critical effects [13].

"Oregonator" and 'Ъrussеlаtоr" studied in detail Ьу the Prigogine school were nevertheless extremely speculative schemes. А study of the behaviour of classical chemical kinetics equations assumed а high priority in order to select the structure responsible for the appearance of critical effects. The results of such а study, described in Chap. 3, сап Ье applied to interpret critical effect experiments.

References рр. 5-6

4

For linear mechanisms we have obtained structurized forms of steady­state kinetic equations (Chap. 4). These forms make possible а rapid deriva­tion of steady-state kinetic equations оп the basis of а reaction scheme without laborious intermediate calculations. The advantage of these forms is, however, not so much in the simplicity of derivation as in the fact that, оп their basis, various physico-chemical conclusions сап Ье drawn, in par­ticular those concerning the relation between the characteristics of detailed mechanisms and the observable kinetic parameters. Ап interesting and important property of the structurized forms is that they vividly show in what way а complex chemical reaction is assembled from simple опеа. Thus, for а single-route linear mechanism, the numerator of а steady-state kinetic equation always corresponds to the kinetic law of the overall reaction as if it were simple and obeyed the law of mass action. This type of numerator is absolutely independent of the number of steps (а thousand, а million) in­volved in а single-route mechanism. The denominator, however, charac­terizes the "non-elementary character" accounting for the retardation of the complex catalytic reaction Ьу the initial substances and products.

In Chap. 5 we present the results of our detailed study of steady- and non-steady-state characteristics for typical non-linear mechanisms and ki­netic models. It appears that the known adsorption mechanism, i.e. the Langmuir-Hinshelwood mechanism, is the simplest опе ensuring multiple steady states оп the catalyst surface C'catalytic trigger"). If this mechanism is completed with а 'Ъuffеr" step, it would Ье the simplest mechanism providing self-oscillations of the reaction rate ("catalytic oscillator"). Itis advisable that approaches under development are applied for the interpreta­tion of concrete catalytic reactions and primarily of the oxidation of simple molecules (Н2 , СО) оп metals. The investigations Ьу Langmuir in the period 1910-1930 initiated the kinetics of heterogeneous catalysis. 80 far these reactions*, demonstrating ап ever-increasing number of new non-trivial properties, have remained а traditional subject of investigation, as сап Ье judged Ьу the number of publications in the Journal о! Catalysis, Kinetica i Kataliz, and Surface Science. It is for this reason that in Chap. 6 we present а detailed analysis of the kinetic model for the reaction of oxidation of СО оп the Group VHI metals. Qualitative and numerical analyses reveal а great variety of steady-state and relaxational characteristics. The constructed kinetic models describe experimental data obtained under both deep vacuum (оп Pt, Ir metals) and normal conditions (Pd-containing catalysts).

We have carried out the analysis of slow relaxations when the times to achieve steady states are much higher than those of separate reactions. It has Ьееп shown that they are caused not only Ьу slow secondary processes but also Ьу the closeness to the region of critical conditions. The general theory of slow relaxations permits us to interpret various peculiarities of

*These reactions are known as the Мопа Lisa of heterogeneous c~talysis.

5

relaxation regimes found in the experiments, e.g. induction periods, "тет­ory" effects, etc. We believe the theory described in Chap. 7 will Ье of interest for the mathematicians who deal with physico-chemical problems.

It is surprising that complicated dynamic behaviour proved to Ье charac­teristic of the simplest and quite ordinary kinetic models of catalytic reac­tions, namely of the Langmuir-Hinshelwood adsorption mechanism. W е are possibly at the initial stage of interpreting the kinetics of complex reactions and the "Sturm und Drang" period has not yet Ьееп completed.

Thus we have described the contents of our book. But what has given impetus to its writing?

First, it is of соттоп interest to unsteady processes and their models. Chemical unsteadiness must Ье taken into account in тапу савев. For example, studies with variations in catalyst activity, calculations of flui­dized catalyst bed processes (when the catalyst grain "is shaking" in а flow of the reaction mixture and has по time to attain its steady state), analyses ofrelaxational non-stationary processes and problems of control. Unsteady state technology is currently under development [14,15], i.e. the technology involving programmed variation of the process parameters (temperature, flow rate, concentration). The development of this technology is impossible without distinct interpretation of the unsteady reaction behaviour.

Secondly, it is necessary to interpret critical effects recently discovered experimentally and which are of соттоп interest. In the adjacent field, i.e. homogeneous catalysis, а great number of such facts have Ьееп accumulated for Belousov-Zhabotinskii reactions. These facts сап Ье interpreted only in terms of the non-linear unsteady-state models.

Thirdly, it is the development of the theory of differential equations that provided chemical kinetics with а new powerful apparatus [16] to Ье put into operation. This apparatus is not only а convenient formal теапв. It will also Ье а Ьаве for а meaningful conceptional language.

The Soviet school of chemical kinetics has accumulated а unique experi­епсе in interpreting concrete catalytic reactions in terms of the stepwise mechanism concept. In the present book we have made ап attempt to inter­pret this experience оп the basis of modern formal kinetics of complex reactions. Since the authors have addressed the book to chemists and mathe­maticians, it is desirable that they both read the whole of the book.

References

1 М. Giua, Storia Della Chimia, Union Tipografico-Editrice Torinese, Torino, 1962. 2 D.A. Frank-Kamenetskii, Diffusion and Heat Transfer in Chemical Кinetics, Nauka,

Moscow, 1967 (in Russian). 3 G.K Boreskov, M.G. Slinko and A.G. Filippova, Dokl. Akad. Nauk SSSR, 92 (1953) 353. 4 W. Davies, Philos. Mag., 17 (1934) 233; 19 (1935) 309. 5 V.V. Barelko and A.G. Merzhanov, Probl. Кinet. Katal., 17 (1973) 182. 6 А.М. Zhabotinskii, Concentration Self-oscillations, Nauka, Moscow, 1974 (in Russian).

6

7 Р. Hugo and М. Jakubith, Chem.-Ing.-Tech., 44 (1972) 383. 8 Н. Beusch, Р. Fieguth and Е. Wicke, Chem. Ing. Tech., 44 (1972) 445. 9 V.D. Be!jaev, М.М. S!inko, V.r. Timoshenko and M.G. S!inko, Кinet. Katal., 14 (1973) 810.

10 M.G. S!inko and М.М. S!inko, Cata!. Rev. Sci. Eng., 17 (1978) 119. 11 Р. Glansdorff and 1. Prigogine, Тhermodynamic Theory of Structure, Stabi!ity and F!uc­

tuations, Wi!ey-Interscience, London, 1971. 12 G. Nico!is and 1. Prigogine, Se!f-organization in Non-equi!ibrium Systems, Wi!ey-Inter-

science, New York, 1977. 13 У а.В. Zeldovich, Zh. Tech. Fiz., 11 (1941) 493. 14 А. Baiker and W. Richarz, Chimia, 30 (1976) 502. 15 Yu.Sh. Matros, Unsteady Processes in Catalytic Reactors, Studies in Surface Science and

Catalysis, Vol. 22, Elsevier, Amsterdam, 1985. 16 A.I. Vo!pert, Qualitative Methods for Chemica! Кirietics Equations, Institut Khimicheskoi

Fiziki, Akad. Nauk SSSR, Chernogo!ovka, 1976 (in Russian).

Chapter 1

Minimum Minimorum

1. Introduction

It is well established that most chemical reactions are complex in nature. For example, the oxidation of hydrogen described ав

2Н2 + 02 = 2Н2 О

but, according to the modern conceptions, а detailed mechanism of the reaction is [1]

(1) Н2 + 02 <=± 20Н

(2) ОН + Н2 <=± Н2 О + Н

(3) Н + 02 <=± ОН + О

(4) О + Н2 <=± ОН + Н

(5) О + Н2 О <=± 20Н

(6) 2 Н + М <=± Н2 + М

(7) 2 О + М <=± 02 + М

(8) Н + он + м <=± Н2 О + М

(9) 20Н + М <=± Н2 О2 + М

(10) ОН + О + М <=± Н2О + М

(11) Н + 02 + М <=± Н02 + М

(12) Н02 + Н2 <=± Н2О2 + Н

(13) Н02 + Н2 <=± Н2О + ОН

(14) Н02 + Н2 О <=± Н2 О2 + ОН

(15) 2 Н02 <=± Н2 О2 + 02

(16) Н + Н02 <=± 20Н

(17) Н + Н02 <=± Н2О + О

(18) Н + Н02 <=± Н2 + 02

(19) О + Н02 <=± ОН + 02

(20) Н + Н2 О2 <=± Н2 О + ОН

(21) О + Н2 О2 <=± ОН + Н02

(22) Н2 + 02 <=± Н2 О + О

(23) Н2 + 02 + М <=± Н2 О2 + М

(24) ОН + М <=± О + Н + М

(25) Н02 + ОН <=± Н2 О + 02

(26) Н2 + О + М <=± Н2 О + М

(27) О + Н2 О + М <=± Н2О2 + М

(28) О + Н2 О2 <=± Н2 О + 02

(29) Н2 + Н2 О2 <=± 2Н2 О

(30) Н + Н02 + М <=± Н2 О2 + М

where М is а "third body", i.e. any substance, commning with which mole­cules exchange energ)". In this саве the number of steps is 30, but it сап Ье even more! ТЬе question a:rises whether they all exist in reality? ТЬеу do. It Ьав long been known from experiments that hydrogen oxidation takes place through intermediates висЬ ав О, Н, ОН, etc. Their concentration is usually

References рр. 45-46

8

low, but these intermediates are of essential importance for the reaction. Hence the reaction mechanism must include steps involving the participa­tion of these substances.

То exemplify а complex mechanism of а heterogeneous catalytic reaction, we will consider the interaction of methane with steam оп а nickel surface. Stoichiometric equations for this reaction are

At the same time а probable detailed mechanism сап Ье represented Ьу

(1) СН4 + Z <=± ZCH2 + Н2

(2) ZCHz + Н2О <=± ZCHOH + Hz

(3) ZCHOH <=± ZCO + Hz

(4) ZCO <=± Z + СО

(5) Z + Hz О ="= ZO + Hz

(6) ZO + СО ="= Z + С02

Here Z represents а catalyst surface site (active centre). ТЬе two final steps are in equilibrium, designated Ьу the symbol ="=. "ТЬе natural classification" ofsimple (elementary) reactions Ьу the number ofmolecules involved simul­taneously in the reaction belongs to Van't Hoff. Ifthe reaction involves one molecule (reaction А --+ В), it is classified as first-order (monomolecular). In cases where two molecules take part in the reaction (e.g. 2А --+ В or А + В --+ С), the reaction is said to Ье second-order (bimolecular). With the participation of three molecules (3 А --+ В or 2 А + В --+ С), the reaction is specified as third-order (termolecular). ТЬе simultaneous interaction of more than three reactants is believed to Ье highly improbable.

ТЬе basic concept in chemical kinetics is the reaction rate. For elemen­tary reactions it is the number of elementary acts in а chemical conversion per unit reaction volume or per unit reaction surface for unit time.

Any chemical reaction сап Ье written in the general form as

NA NB

I lXiAi <=± ~ /3i Bi (1)

where IX; and /3; are the stoichiometric coefficients for the initial substances А; and reaction products Bi , respectively, and NA and NB are the number of initial substances and reaction products, respectively.

For simple reactions the аЬоуе values are limited. Thus, IX; and /3; сап only Ье equal to О, 1, 2 or 3 (rarely).

Stoichiometric equations are free from these limitations. For example, the С2Н4О oxidation reaction is

9

(2)

Reactions in which the initial substance is simultaneously its product are rather rare. These "propagation" reactions were called auto-catalytic steps Ьу Kondratiev and Nikitin. For example, one of the steps in the thermal dissociation of hydrogen is

Н2 + н <=± Н + Н + Н (3)

Ав а rule, in simple reactions the initial substances and products are different and the reaction does not yield any of the initial substances.

In the саве of gas-phase catalytic reactions, the reactants are, оп the one hand, gaseous and, оп the other hand, surface substances. The latter are оп the surface of solid catalysts. А simple catalytic reaction is described ав

NA Nx NB Ny

I aj Aj + I a;Xj <=± I f3j Bj + If3;Yj (4) i=l i=l i=l i=l

where Aj and B j are the initial substances and products in the gas phase, respectively, aj and f3j are their stoichiometric coefficients, Xj and Yj are the surface substances, and а; and fЗ; are their stoichiometric coefficients.

Ав usual, eqn. (4) is of the form

NX N y

аА + I a;Xj <=± fЗВ + I f3;Yj (5) i=l i=l

Here а and f3 = 1 or О, i.e. it is assumed that а simple catalytic reaction either involves the participation of only one molecule of gaseous substance (e.g. СН4 + Z <=± ZCH2 + Н2 ) or it proceeds without the participation of these substances (e.g. ZCHOH <=± ZCO + Н2 ). The stoichiometric соет­cients а; and fЗ; are assumed to equal 1, 2, or 3 (rarely) and La; and Lf3; ~ 3.

The rate of а simple reaction ав represented Ьу eqns. (1) and (4) is deter­mined from the difference between the rates of the direct and reverse reac­tions

(6)

Under equilibrium, w = О and ш+ = ш-. The dependence of the rates for the direct and reverse reactions оп the

concentration ofthe reactants is expressed in terms ofthe law ofmass action ав

(7)

w (8)

References рр. 45-46

10

where СА and СВ are the concentrations of the initial substances and products, 'respecti~ely and k+ and k- are rate coefficients for the direct and reverse reactions to determine а specific reaction rate, i.e. the rate when the concentrations of reactants are equal to unity. They fit the Arrhenius rela­tionship and increase exponentially with temperature

k; ехр (- Е+ jRT)

k(j ехр (- Е- jRT)

(9)

(10)

Here k; and k(j are the pre-exponential factors, Е + and Е- are the activation energies for the direct and reverse reactions, respectively, R is the universal gas constant, and Т is the absolute temperature.

The ratio of the rate coefficients for the direct and reverse reactions сап Ье used to define the equilibrium constant

k+ K eq = k-

Homogeneous catalytic reactions fit the relationship

w = ш+ _ ш- = __ 1_ dNAi = _1_ dNBi

cxi V dt fЗi V dt

(11)

(12a)

Here NA and NB are the concentrations of the substances in the system and V is the' volum~ of the system.

Heterogeneous catalytic reactions fit the relationship

= ш+ _ ш- = __ 1_. dNAi 1 dNBi

W СХ; Scat dt = fЗj Scat • dt (12Ь)

Here Scat is the surface area of the catalyst for conversion. For steps taking place without changing the number of moles, eqns. (12а)

and (12Ь) take the usual form

+ _ 1 dCAi W = W - W = - ---

СХ; dt (13)

The reactions whose rates fit the mass action law сап Ье called simple. This postulate is, however, inaccurate. Опе сап meet complex reactions involv­ing several steps, but which nevertheless fit the above law. According to Laidler, simple (elementary) reactions take place Ьу overcoming one energy barrier (опе barrier per elementary reaction).

Strange as it seems, there are one-step reactions in nature, e.g. first-order reactions of monomolecular decomposition

C2 H5 Br -> С2Н4 + HBr

and

the second-order reactions

2NOI -+ 2NO + 12

and

со + 02 -+ С02 + О

and the third-order reaction

2NO + 02 -+ 2N02

11

These савев are, however, exceptions rather than the rule. Most reactions are now considered to Ье multi-step and complex.

2. Chemical kinetics and linear algebra

А natural language accounting for the stoichiometry of chemcial reac­tions is that of linear algebra. Let ив remind ourselves of its basic concepts. А matrix is а rectangular array of numerals containing т rows and n

columns. А general form of this m-Ьу-n matrix is

г a j2 ... а,,. ]

А а2! а22 . " а2n

аm! аm2 аmn

Here аи are matrix elements and i is the row number and j the column number whose intercept provides а place for the element a ij . If т = 1, n > 1, а one-row matrix [a j , а2 , ••• аn ] is obtained. ТЫв is called а vector row. For т > 1 and n = 1 we have а one-column matrix called а vector column

Matrices сап Ье added and multiplied Ьу воте numerals and Ьу each other. The latter operations are often used in formal chemical kinetics. The sum of two m-Ьу-n matrices А = аи and В = Ьи gives а new m-Ьу-n matrix, С = Си, whose elements are determined Ьу the equality Cij = аи + bij • The sum of matrices is designated ав С = А + В. А new matrix kA = [kaij] is the product of multiplying matrix А = [aij ]

Ьу k. For this rriultiplication it is necessary to multiply every element of the matrix Ьу k.

Let А = [ац] Ье ап m-Ьу-n matrix and В = [Ьц] Ье ап n-Ьу-р matrix, i.e.

References рр. 45-46

12

the number of rows in matrix В is the same as the nurnber of columns in matrix А. Then the product ofmatrices А and В (designated as АВ) is called ап m-Ьу-n matrix, С = [Сц), whose elements are determined Ьу the equalities

(14)

where i = 1, 2, ... , т; j = 1, 2, ... , р The element Ci,j of matrix С, positioned at the intercept of the ith row and

the jth column, is equal to the scalar product of the ith vector-row of matrix А and the jth vector colurnn of matrix В.

Опе must remember that the product АВ is defined only if the number of columns of matrix А is езuа1 to the number of rows of matrix В. If the ro~f' and columns in matrix А interchange their places, it produces matrix А , which is called а matrix transposed with respect to matrix А

г а21 "., ]

АТ а12 а22 аm2

a1n а2n аmn

Ifthe number ofrows and columns is the same, the matrix is called square. This nurnber is called а matrix order. The totality of elements (a1l, а22 , •••

аnт ) of the square matrix is its main diagonal. Each square matrix is assurned to correspond to а certain value (to Ье

more precise, to а numerical function) which is called а matrix determinant. For а first-order matrix, i.e. the number of aij , the determinant is equal to this number itself

(15)

А second-order determinant is equal to

(16)

А minor for element аи of matrix А is the determinant for the matrix obtained from matrix А with the ith row and jth column omitted. Ап alge­braic complement for element аи of matrix А is its minor multiplied Ьу (- 1Y+j. Matrix determinant amounts to the sum of the products of elements for some column (or row) with their algebraic complements. Ап important concept is the matrix rank. The highest order of the поп­

zero determinant generated Ьу а given matrix is designated as rgA. With а zero determinant the matrix is singular. The matrix whose elements along the main diagonal are equal to unity and аll the rest are zero is called а unit or identity matrix. If some vector У is а linear combination of vectors, it is called linear-dependent

13

(17)

The rank for а set of vectors is the maximum number of linearly indepen­dent vectors. If, from the total set of vectors, one chooses the combination containing the maximum number of linearly independent vectors it will Ье а basis. The matrix rank remains unchanged if one adds а row that is а linear combination of the other rows and if this row is cancelled.

It is very convenient to apply а matrix technique to solve sets of linear equations. The set of linear equatiqns relative to the n unknowns Х1 , Х2 ,

... , хn is

{""Х, + а12 Х2 + + а1n хn = Ь 1

а21 Х1 + а22 Х2 + + а2n хn = Ь2 (18)

аm1 Х1 + a m2 x z + ... + аmnхn = Ьm

In matrix representation it is

Ах = в (19)

If В = О, the system is called homogeneous. But what is the field for the direct application of linear algebra concepts

in chemical kinetics? The chemical composition of substances is accounted for Ьу the molecular

matrix whose element aij is the atomic number of the ith element entering into the jth reactant molecule. Thus for the mixture of С, 02' СО and С02 , the molecular matrix is

с о

(20)

The rows here correspond to the substances С, 02' СО and С02 and the columns are the elements С and О.

Molecular masses of substances are determined Ьу the equation

(21)

Here М is the vector column of the molecular masses of the substances, А is the molecular matrix and МА is the vector column of the atomic masses. The size of matrix А is N-by-m, where N is the number of reactants in the system and т the number of chemical elements entering into the composi­tion of reactants.

References рр. 45--46

14

А stoichiometric matrix is опе whose elements are the stoichiometric coefficients of the reacting substances. Its rows correspond to the reactions and its columns are the reacting substances.

For the reactions

(1) 2 С + 02 = 2СО

(2) 2СО + 02 = 2С02 (22)

(3) С + 02 = С02

the stoichiometric matrix Г is of the form

С 02 СО С02

Г~ -1 2 О

] 1

Г -1 -2 2 2

-1 -1 О 1 3

(23)

The stoichiometric coefficients of the initial substances for а given reac­tion are included with а negative sign. Those of the reaction products are positive. If а substance does not participate in а reaction, its stoichiometric coefficient is taken to Ье zero.

The size of а stoichiometric matrix is (s-by-N), where s is the number of reactions and N the number of reactants in the system. Stoichiometric equations for а complex reaction сап Ье represented as

Га = () (24)

Here а is the vector column of the reactants. Thus, the stoichiometric equations (22) сап Ье obtained Ьу multiplication

[-: -1

-1

-1

-1 {

- 2 С - 02 + 2 СО = О

- 02 - 2 СО + 2 С02 = О

- С - 02 + С02 = О

What requirements must the stoichiometric matrix Г fit? Chemists choose stoichiometric coefficients such that, in each reaction, the number of atoms оп the left-hand and right-hand sides are the saтe for every element. Непсе the law of constant mавв for atoms of а given type must hold over the reaction steps. In matrix representation, this requirement is ot' the form

ГА = о (25)

Let ив multiply matrices (23) and (20)

15

[

( - 2) х 1 + (-1) х о + 2 х 1 + О х 1 (- 2) х О + (-1) х 2 + 2 х 1 + О х 2]

~ О х 1 + (-1) х О + (- 2) х 1 + 2. х 1 О х О + (- 1) х 2 + (- 2) х 1 + 2 х 2 ~ О

( - 1) х 1 + (- 1) х О + О х 1 + 1 х 1 ( - 1) х О + (- 1) х 2 + О х 1 + 1 х 2

We obtain а matrix in which аН the elements are zero (а zero matrix). Непсе the stoichiometric matrix is written correctly.

The size ofthe matrix ГА obtained Ьу multiplying matrices Г (s·by-N) and А (N-by-m) is (s-by-m). If this matrix is multiplied Ьу the vector column of atomic таввев МА then, taking into account eqn. (21), we obtain

(26)

Ттв is the equation for the law of mass conservation over steps. Equation (25) makes it possible to construct correctly the stoichiometric

matrix Г for а given number of substances and, hence, molecular matrix А. Among the rows of this matrix опе сап find those that are linearly depen­dent. Thus, in matrix (23), the third row will Ье obtained if опе adds the two upper rows and multiplies the вит Ьу 1/2.

The rank of matrix Г сап never Ье аЬоуе N - т, where т is the number of chemical elements in the system. This holds due to the fact that there always exist т linearly dependent columns of matrix Г set Ьу eqn. (25)

rgf ~ N-m

The rank of matrix А is

rgA = min (N, т)

(27)

(28)

Ав а rule, rgA ав

т, then the rank for а stoichiometric matrix is determined

rg Г ~ N - rgA (29)

This relationship is caHed the Gibbs stoichiometric rule. For example, for the mixture ofreactants С, 02' СО, С02 , N = 4, rgA = 2

[вее eqn. (20)]. Then rgf ~ 2; hence in matrix (23) апу two rows are linearly independent.

Substances corresponding to the linearly independent columns of matrix Г are caHed the key substances; the remainder are referred to ав non-key substances. It is evident that the number of key subst.ances is equal to the

References рр. 45-46

16

rank of а stoichiometric matrix. For the С + 02 + СО + С02 mixture опе сап сЬоове апу two substances to Ье the key substances.

In the general саве, the solution of the problem of constructing а stoi­chiometric matrix using eqn. (25) is ambiguous. In principle, опе сап obtain multiple sets of stoichiometric equations involving those that will Ье linear­ly dependent. If stoichiometry is expected to Ье more than а simple 'Ъооk­keeping" of material components in а chemical system it is necessary to impose воте physico-chemical restrictions, primarily оп the number of reacting substances in еасЬ elementary reaction (it must Ье ~ З). ТЬеп the number of possible variants is sh~rply reduced. 80 far several computation algorithms to construct stoichiometric matrices have Ьееп developed. ТЬе computer derives chemical equations taking into account the phY8icallimi­tations specified. It is evident that these algorithms will Ье effective to describe chemical conversions in complex multi-component mixtures. In simple савев their application would Ье senseless.

3. Unsteady- and steady-state kinetic models

ТЬе kinetics of chemical reactions i8 studied in laboratory reactors. ТЬеве reactors сап Ье divided into several groups, taking into account whether they belong to closed or ореп sY8tems. Closed systems сап exchange energy with the environment but do not exchange substance. Ореп systems ех­change either substance and energy or only substance. А laboratory catalytic reactor that i8 а closed system, is said to Ье а 8tatic

system. ТЬе kinetic model for complex reactions taking place in the reactor is of the form

с de гТ ш(е) (ЗО) - = dt

~T

Here eis the vector column of substance concentrations, Г is the transposed stoichiometric matrix, and ш(е) is the vector column of reaction rates deter-mined from eqns. (6}-(8). It is ап unsteady-state kinetic model.

Let ив write висЬ а model for the catalytic reaction of methane with steam оп а nickel catalY8t. Its detailed mechanism Ьав Ьееп given above.

The stoichiometric matrix is

СН. Н2 О Н2 СО СО2 Z ZCH2 ZCHOH ZCO ZO

-1 О 1 О О -1 1 О О О 1

О -1 1 О О О -1 1 О О 2

f о о 1 О О О О -1 О 3

О О О 1 О 1 О О -1 О 4

О -1 1 О О -1 О О О 1 5

О О О -1 1 О О О -1 J 6

17

The transposed matrix is

-1 О О О О О

О -1 О О -1 О

1 1 1 О 1 О

О О О 1 О -1

-ГТ О О О О О 1

-1 о О 1 -1 1

1 -1 О О О О

О 1 -1 О О О

О О 1 -1 О О

О О О О 1 -1

and the vector column of reaction rates is

Ш]

Ш2

W Wз

Ш4

Ш5

Ш6

~T . After multiplying matrix Г Ьу the vector column of the reaction rates, ш, we obtain ап unsteady-state kinetic model

d[CH4 ] =

d[H2 O] = -d-t- -ш] dt

- Ш2 - Ш5

d[H2 ] Ш] + Ш2 + Wз + Ш5

d[CO] = dt dt

Ш4 - Ш6

d[C02 ] d[Z] - Ш] + Ш4 - Ш5 + Ш6 = Ш6 dt dt

d[ZCH2 ] =

d[ZCHOИJ =

dt Ш1 - Ш2 dt

Ш2 - Wз

d[ZCO] =

d[ZO] dt

Wз - Ш4 dt Ш5 - Ш6

For а closed system the laws ofmass conservation for various atoms must Ье met

References рр. 45-46

18

(31)

where А is the molecular matrix; с the vector column of substance соп­centrations, and С the vector column for the numbers of the different atoms.

Thus, for the Н2 + 02 + Н2 О mixture

н о

[: :] Г] А [~ ОГ ~T О А с Со,

2 02 2

1 Н2О Сн2о

АТс С [2 О 2] Г: ] ~ [2c~ + 2c"~ ] х

О 2 1 2С02 + СН20 Сн2 о

Consequently, СН = 2СН2 + 2сн2о, СО = 2С02 + СН20' and СН and СО = total amounts of О and Н atoms in the system. It is evident that the equations of type (31) must fit апу stoichiometry of complex reactions.

The most popular catalytic reactor which is ап ореп system is а gradient­less apparatus. The principle of its operation is to равв а reaction mixture through а system with а catalyst under intense stirring. In this саве the врасе velocity is lower than the stirring rate. А mathematical model for the unsteady-state heterogeneous catalytic

reaction in а gradientless reactor is

~[ = v;,f(e, х) + vоео - vе

х = g(e, х)

(32)

(33)

where с is the vector of substance concentrations in the gas phase, х the vector of substance concentrations оп the catalyst surface, [(е, х) and g(e, х) the vector functions ofkinetic relationships derived according to the surface action law, vc: and ~ the volumes of the catalyst and the gas phase in the reactor, respectively, Vо and v the inlet and outlet flow rates of the reaction mixture, and со the vector ofthe inlet concentrations ofreactants in the gas phase. For а closed catalytic system, Vо = v = О. Equations (32) and (33) represent the unsteady-state model [ог а continuously stirred tank reactor.

Оп the basis oftheir heterogeneity, reactants are divided into two groups: gas-phase substances having concentration с and surface substances with concentration х.

It is possible to classify substances in the other ways. Thus, опе сап discriminate "оЬвегуаЫе" and "unobservable" substances depending оп whether their concentration is ог is not controlled in the experiment. Then substances аге classified ав "гетоуаЫе" and "поц-гетоуаЫе" [гот the system when exchanging with the environment and, finally, initial substan-

19

ces and products оп the one hand and intermediates оп the other. The latter do not enter into the stoichiometric equations of complex reactions. The аЬоуе classifications, however, depend to а considerable extent оп ех­perimental techniques. As far as the final stoichiometric equation is con­cerned, their choice is not ideal. After аН one сап arrive at а simple result: substances сап Ье present both in the gas phase and оп the catalyst surface. In most cases the latter are "unobservable" and "non-removable" and it is these substances that are not recommended to Ье included in the final reaction equations.

Concentrations of gaseous substances must satisfy the state equation

N Р

.2:: С; = RT t=l

(34)

where Р is the pressure of the gas mixture, Т the absolute temperature, and R the universal gas constant.

The concentrations of the surface substances are related Ьу the equation n

2:: bijXi = Cj i=l

(35)

Here bij = the number ofj-type active sites entering into the ith substance. In most cases it is assumed that there exists only one type of active site, hence the linear law of conservation, eqn. (35), is unique.

Ав а rule Ьи = 1. For the reaction ofmethane with water vapour, eqn. (35) сап Ье rewritten ав

Xz + XZCH2 + XZCHOH + xzco + xZ9 = Cz

(here Cz = 1014_1015 molecules сm -2). Now if the concept of surface coverage, i.e. dimensionless concentrations

0 ; = Xi/C" is introduced, we obtain n

2:: 0 ; = 1 i=l

provided that there is only one type of active site and b;j = 1. After achieving the steady state, eqns. (32) and (33) transform into

v;,f(c, х) + vосо - vс = О

g(c, х) = о

(36)

(37)

The steady state for а closed system is equilibrium. From eqn. (37) we obtain

vс - uосо v;, = f(c, х) = w (38)

Equations (36) and (37) represent the steady-state model. After achieving

References рр. 45-46

20

the steady-state, eqn. (38) is used to determine W, the steady-state rate for concentration variations of gas-phase substances, experimentally.

If опе suggests that, in the course of the experiment, the flow rate of the reaction mixture varies insignificantly, и ~ иа , then eqns. (32) and (38) сап Ье written ав

Ygt = V';,f(c, х) + и(Са - с) (39)

и(са; с) = f(c, х) = ИТ. с

(40)

Опе must note the significant peculiarity of mechanisms and models for catalytic reactions. ТЬе sequence of steps in the detailed mechanism always contains at least опе cycle of intermediate substances, i.e. а catalytic cycle. It теапв that substances containing active catalytic sites are both consumed and formed. Тшв сап Ье confirmed Ьу the stoichiometric matrix for the reaction of methane with water vapour.

4. Steady-state reaction theory

Most catalytic reactions take place under steady conditions. In the 1950в and 1960в Horiuti and Nakamura [2] and Temkin [3] developed а steady-state reaction theory. Reactions are calIed steady-state provided that, with соп­stant concentrations of gas-phase substances, the concentrations of substan­сев оп the catalyst surface are aIso unchanged. Ап example is provided Ьу the conditions observed in а gradientless reactor. А mathematical modeI for а steady-state reaction is given Ьу the aIgebraic set of equations (36) and (37). Ofimportance is the concept ofthe pseudo-steady-state hypothesis. QuaIita­tively it is specified ав "matching" intermediates to а reactor gas mixture. With varying concentrations of reactants in the gas phase, the concentra­tions of intermediates change во that, in practice, they do not differ from the values corresponding to the steady reaction conditions with constant соп­centrations of gaseous substances amounting to given instantaneous соп­centrations. In this саве а mathematical reaction modeI takes the form of the аIgеЬгаiс-diffегепtiаl set

Ygt" = V';,f(c, х) + иаСа - ис

g(c, х) = о

(41)

(42)

Subset (42) treated under the assumption of constant concentrations of gas-phase substances is the set of pseudo-steady-state equations

g(x) = о (43)

Pseudo-steadiness is опе of the most essentiaI characteristics for catalytic reactions. In what folIows this concept is specified more rigorously оп the

21

basis of the differential equation theory. It is necessary Ьесаиве studying quasi-steady-state equations (the method of quasi-steady-state concentra­tions) is the most popular approach in the formal kinetics of complex reac­tions.

For the steady-state reaction theory it is not important whether the reaction is steady or pseudo-steady. Only the fact that the formation and consumption rates of intermediates are equal is of importance.

Horinti has introduced the concepts of "independent intermediates", "stoichiometric number", "reactiQn route" and "independent reaction routes" that Ьауе Ьееп extensively used in the steady-state reaction theory. Let ив clarify them Ьу а model izomerization reaction with а detailed mе­chanism

А + Z <=± AZ 1

AZ <=± BZ 1 (44)

BZ <=± В + Z 1

In this саве there are three intermediates AZ, BZ, and Z, the latter being ап active catalytic site. Their surface coverages are related Ьу at least опе balanced relationship

E>z + Е> AZ + E>BZ = 1 (45)

Ав а rule, this relationship is unique. According to eqn. (45), the number of linearly independent intermediates, 1, will Ье lower then their total number, 1tot , Ьу unity

(46)

То obtain stoichiometric equations for the reaction steps (brutto-equa­tions) that do not involve intermediates, опе must add the steps of the detailed mechanism, first multiplying them Ьу the numbers specified. For the simple mechanism of eqns. (44) these numbers equal unity and are placed оп the right-hand side of the equations of the reaction steps. Horiuti specified these numbers ав stoichiometric numbers (not to confuse them with stoi­chiometric coefficients, which indicate the number ofmolecules of а reacting substance). Stoichiometric numbers must fit the equation

(47)

Here "Т is the transposed matrix of stoichiometric numbers and f int is the matrix of stoichiometric coefficients for intermediates. Elements of the latter are taken to Ье negative if substance is consumed in а given reaction step, positive if it is formed, and zero if substance is not involved in the reaction step. Multiplication of matrix "Т (P-by-s) Ьу matrix f int (s-by-1tot ) gives the matrix VTtint whose size is (Р-Ьу-1tot ) (s is the number of steps).

The vector column for the matrix of stoichiometric numbers ji (s-by-P) is called the route of а complex reaction. ТЬе rank of matrix f int cannot Ье

References рр. 45--46

22

аЬоуе (8 - Р) where Р is the number of linearly independent routes. It does take place due to the fact that, according to ечп. (47), there are Р linearly dependent rows and usually

rgI\nt = 8 - Р (48)

Оп the other hand, with а unique law of conservation with respect to catalyst

rgf\nt = 1 = I tot - 1

After substituting ечп. (49) in ечп. (48), we obtain

Р = 8-Itot +1

(49)

(50)

This relationship to determine the number of linearly independent routes is called Horinti's stoichiometric rule.

For isomerization reaction (44), 1 = 3 - 1 = 2, hence Р = 1. This reac­tion is а one-route reaction.

Let us now consider а more complex саве. А detailed mechanism for the synthesis of vinyl chloride оп а mercuric chloride + medical charcoal catalyst сап Ье represented Ьу the sequence of steps

I П

HgCl2 ' HCl + С2Н2 +z HgCl2 • С2Н2 • HCl 1 О

HgCl2 ' HCl + HCl <=t HgCl2 ' 2HCl О 1 (51)

HgCl2 ' С2Н2 • HCl + HCl ---+ HgCl2 ' HCl + C2 HsCl 1 О

HgCl2 '2HCl + С2Н2 ---+ HgCl2 'HCl + С2Нз Сl О 1 С2Н2 + HCl = C2 HsCl

For this mechanism the initial catalytic centre is HgCl2 ' HCl and the surface coverage of coal Ьу pure mercuric chloride is neglected. Here we have three intermediates. ТЬеу are HgC12 ' HCl, HgCl2 ' 2HCl and HgCl2 ' С2Н2 • HCl (8 = 4, Р = 4 - 3 + 1 = 2). Consequently, two indepen­dent reaction routes are observed. Vector columns of stoichiometric пuш­bers are given to the right of the step equations.

ЕасЬ step of the mechanisms detailed аЬоуе involves по more than опе molecule of воте intermediate. Such mechanisms are called linear.

Let us now consider ап example of а non-linear mechanism, including а reaction that involves two molecules of some intermediates. ТЬе probable reaction mechanism for ammonia synthesis оп ап iron catalyst сап Ье represented as

Z + N2 +z ZN2 1

ZN2 + Н2 +z ZN2И2 1

1 (52)

ZNH + Н2 +z Z + NНз 2

N2 + 3Н2 = 2NНз

23

8tep 3 is non-linear. There are four intermediates in this reaction, р = 4 - 4 + 1 = 1. In this саве, unlike the one-route reaction of eqn. (44), the stoichiometric numbers are not аН equal. Routes are vectors and they сап Ье treated with the вате operations ав vectors. ТЬеу сап Ье multiplied Ьу arbitrary numbers. If vector а (1;1;1), which is а route for the detailed mechanism (44) and leads to the st6ichiometric equation А = В, is multi­plied Ьу an arbitrary number т, the vector та is obtained. ТЫв vector is also а reaction т А = тВ.

It is evident that the choice of stoichiometric numbers is ambiguous. In principle, one сап obtain an infinitive number of routes Ьу obtaining them ав linear combinations of routes entering into the basic route.

But what are the real routes of а chemical reaction? А mechanism for а complex reaction is described completely Ьу its set of steps, whereas its set of routes cannot Ье chosen adequately. It is of importance to understand that reaction steps are real, whereas reaction routes are one of the ways to represent chemical 'Ъооk-kеерiпg".

8teady-state reaction theory gives an answer to an important question, namely that if one knows а kinetic law for воте elementary reaction, in what way сап an equation for the complex reaction rate Ье derived?

Let ив introduce а concept ofreaction run over а given route [4]. Reaction run is determined ав the disappearance ofmolecules ofthe initial substances and the appearance of those of the products whose amount is determined Ьу the coefficients of the stoichiometric equation corresponding to а given route. Reaction rate V(р) over the route Р is specified ав the number of the respective runs per unit time or unit surface. ТЬе rates of individual steps, ш+ and ш- , are always positive, whereas V(р) сап also Ье negative depending оп the stoichiometric chemical equation.

Let ив write the steady-state condition for the steps

(53)

Here V(р) is the vector column ofthe rates over the reaction routes and w the vector column for the rates ofthe steps. 80 the rate of every step is represent­ed ав а linear combination of the rates over the reaction routes.

ТЬе sense of condition (53) reduces to the fact that the formation rate of intermediates is equal to their consumption rate. For unit time in а unit volume or оп а unit surface there form r;ntW molecules for an intermediate. Note that the Г~' matrix size is C4ot-by-s). After substituting W from eqn. (53), we obtain

(54) ~T

According to eqn. (47), r int V = о. Hence rintw о also, ав was to Ье shown.

References рр. 45--46

24

Let us derive the basic equation for the steady-state reaction theory [3J. For this purpose we will use the identity

(Ш 1 - W_ 1 )W2 Wз · .. + W_ 1 (W2 - W_ 2 )Wз •·• + W_ 1 W_ 2 (Wз - W- з)··· + ... = W1 W2 WЗ ••. - W_ 1 W_ 2 W_ З . . . (55)

Expressing (Ш1 - w_ 1 ), (Ш2 - Ш_2), etc., Ьу the equation

gives

ШЭ - w- s

р

I v~P)v(p) p~1

(V\I)V(l) + V(2)V(2) + .. . )W2 Wз . .. + W_1MI)V(l) + V~)V(2) + .. . )Wз . •. + + w_ 1 w_ 2 MI)V(l) + V~2)V(2) + ... ) ... + ...

W1 W2 WЗ · .• + W_ 1 W_ 2 W_ З · ..

Dividing both sides of the аЬоуе equality Ьу Ш1 Ш2 Wз ... and rearranging the terms gives

... )

= 1 W_ 1 W_ 2 W_ З · ..

W 1W 2W 2 · .. (56)

It is this equation that is the basic equation for the steady-state reaction theory. It holds irrespective of the order of numbering the steps, which is confirmed Ьу

= 1 _ _ w ___ s.:-' _w_-_S..:.2 _W_-...:S3_·_·_· -' •

WS1

WS2 WSз WЭ4 •••••

+ ... )

Here the steps are numbered not Ьу the numerals 1, 2, etc., but Ьу double indices to show that the sequence of steps сап Ье chosen arbitrarily.

If at least one of the reaction steps is irreversible, the term (ш -s, W -S2 W -Sз .. . !ws, W

S2W

S3 ••• ) оп the right-hand side of eqn. (57) disappears.

То obtain а dependence of the rates over the Р independent routes оп the concentration of gas-phase substances, it is necessary to have а set of s = Р + 1 equations, eqn. (57), which contain 1 independent intermediates entering into the equation for the reaction rates ш• and W_ S '

This technique is not recommended for very simple mechanisms (e.g. linear mechanisms withtwo intermediates). In these cases one should derive а set of pseudo-steady-state equations. It will Ье а set.of algebraic equations

25

that are linear with respect to the intermediates. Their concentrations and Ьепсе the step rates сап easily Ье determined. For really complex reactions (primarily multi-route reactions), опе cannot omit special algorithms simpli­fying the derivation of kinetic equations. А significant simplification of the algorithm is associated with applying

chemical kinetic methods taken from the graphs theory. А graph is а geome­trical всЬете consisting of а set of points connected Ьу lines. It сап Ье а complex electric всЬете, а railway network, а plan of constructional works or finally, а complex chemical reacti"on.

Graph theory ав а mathematical discipline was elaborated Ьу Euler. In 1736 Ье published his article in which Ье proved the insolubility of а problem of keen interest to the citizens of K6nigsberg: whether it is possible to visit all the seven bridges over the River Preigel, crossing еасЬ bridge only опсе. This problem оп the K6nigsberg bridges was unique for more than 100 years. Only in the middle of the 19th century was the interest in the graph theory reawakened. ТЬе famous four colours problem was formulated. In this it is necessary to paint different countries in such а way that neighbours are coloured differently. For а long time the problems оГсоmmivоуаgег", "five queens", "jealous husbands", "Attila's horse", "Hanoi tower", etc. enriched the "science collection" section ofpopular magazines. Only in the last thirty years Ьав graph theory entered а new period: it Ьав Ьееп extensively applied in electrical engineering, economics, biology, psychology and chemical kinetics.

King and Altman [5] and Temkin developed а method to represent а reaction mechanism ав а graph. Its nodes are intermediates and its edges are steps. Reaction directions are marked Ьу arrows оп the edges. А graph corresponding to the detailed mechanism for the isomerization

reaction (44) is represented ав

yZ\1 Lз 1

2 8Z'" .AZ

-2

А detailed mechanism for vinyl chloride synthesis, eqn. (51), is accounted for Ьу the graph

2 1

V -2 ~ ~ -1 ~ HgCl2·2HCL HgCl2·HCl HgCl2·C2H2·HCL

~ /'''--. .-/ 4 З·

References рр. 45-46

26

А graph circ1e is а fina1 sequence of the edges in which по node except the starting point оссшв twice. А graph for the isomerization reaction has one circ1e, whereas that for viny1 chloride synthesis contains two circ1es. Every route of а chemica1 reaction corresponds to а graph's circ1e and vice versa. The number of independent reaction routes is equa1 to the number of е1е­ments in the basis of circ1es. It permits иэ to determine independent reaction routes from the graph type.

Let иэ specify definitions that are necessary for further description. А tree is any sequence of the graph edge.s containing по circ1es. А spanning tree (maxima1 tree) is а sequence of edges going through all the nodes of the initia1 graph. It suffices to add at 1east one edge to the spanning tree so ав to obtain а circ1e. Spanning trees of nodes are specified in this way if they enter this node. Spanning trees characterize routes of conversions due to which а given intermediate is formed from а combination ofthe others. Thus, for an izomerization reaction we have the following spanning trees.

Spanning trees of node Z

L ---s 2 -2

Spanning trees of node AZ

Spanning trees of node BZ

~1 _ А1

-}" "" Consequently the one-route isomerization reaction has n nodes and n steps. It сап easi1y Ье shown that n spanning trees enter in every node and the tota1 number of spanning trees is n 2

• For the mechanism under consideration n = 3 and the overall number of spanning trees will Ье nine. Edge weights are obtained if the rates of steps (direct and reverse) corresponding to the graph's edge are divided Ьу the concentrations of reacting intermediates.

ш+ Ь+

, (58) ,

х+ 1

Ь,- = ш,-

(59) Xi-

Here Ь,+ and Ь; are the edge weights for the direct and reverse steps and Xi+

27

and Xj- the concentrations of the intermediates for the direct and reverse steps, respectively. This characteristic is also called а frequency, а kinetic coefficient or probability. The physico-chemical sense of the edge weight is simple. It is а step rate with unit concentration of reacting intermediate.

Step weights for the isomerization mechanisms are Ь1 k1[A] Ь_ 1 k_ 1

Ь_ 2 = k_ 2 (60)

Ь_ З = k_з[В].

Spanning tree weights are equal to the product of its constituent edge weights. Let ив determine the weights for the spanning trees entering into different nodes.

For node Z

BZ,1 = Ь2 ЬЗ ,

For node AZ

ВЛZ,1 = ЬзЬ j ,

For node BZ Влz,2 Влz,з

BBZ,1 = b1b2, BBZ,2 = Ь2 Ь- з , Ввz,з = Ь_1Ь_ з The overall spanning tree weight of the nodes is

Z: Bz = ЬzЬз + ЬЗЬ- 1 + Ь- 2 Ь- 1

AZ: ВЛZ

BZ: BBZ

ЬЗЬ 1 + b1 b_ 2 + Ь_ З Ь_ 2

b1b2 + Ь2 Ь_ з + Ь_1Ь_ з

The overall spanning tree weight of the graph is

В = Bz + Влz + BBZ

(61)

(62)

(63)

(64)

(65)

Let ив write а formula that will considerably simplify the derivation of kinetic equations for complex reactions

Вх Х = В (66)

Here Х is the intermediate concentration, ВХ the overall weight of node spanning trees corresponding to а given substance and В the total spanning tree weight for а graph.

Equation (66) (Mazon law in electrical engineering) was used [6] to cal­culate steady-state rates of complex fermentation reactions. Recently [7], the correctness of this equation has been confirmed.

With known concentrations of intermediates one сап readily determine the required rate of any step. For а one-route mechanism this rate is just the rate over the route. For example,

(67)

References рр, 45-46

28

Since k2 = Ь2 and k- 2 = Ь_ 2 , eqn. (67) сап Ье rewritten аа

w

ТЬеп

w b,BAZ - b_,BBZ

В

Ь,(ЬзЬ j + bjb_, + Ь_зЬ_,) - b_,(bjb, + Ь,Ь_ З + Ь_jЬ_ З )

В

(68)

Ьj?zЬз - Ь_jЬ_ZЬ_ З (69) bjb, + Ь,ЬЗ + ЬзЬj + b_jb_ z + Ь_zЬ_ з + Ь_зЬ_ j + bJb_, + Ь,Ь_ З + ЬзЬ_ j

Substituting step weights from eqn. (60) into eqn. (69) gives

kjkzkз[А] - k_jk_zk_з[В]

w = k1 [A](k2 + kз + k_ 2 ) + k_з[В](k_ 1 + k_ 2 + k2) + k2kз + k_ 1k_ 2 + kзk_ 1 (70)

Let иа analyze the structure of eqn. (70). Its numerator сап Ье written аа К+ [А] - К- [В], where К+ = k 1 k2 kз and К- = k_ 1 k_ 2 k_ з . In this form, it corresponds to the brutto-equation of the reaction А = В obtained Ьу ad­ding аН the steps of the detailed mechanism with unit stoichiometric пшn­bers. ТЬе numerator is а kinetic equation for the brutto-reaction А = В

considered to Ье elementary and fitting the тавв action law. ТЬе denomina­tor accounts for the "non-elementary character" due to the inhibition ofthe complex catalytic reaction rate Ьу the initial substances and products.

If аН the steps are irreversible (k_ 1 = k_ 2 = k- з = О), eqn. (70) is sim­plified considerably to

w = k1 k2 kз [А] (71) k1 [A](k 2 + kз ) + k2 kз

ТЬе аате algorithm сап also Ье effectively used for complex multi·route reactions. А graph for а linear mechanism of апу complexity сап always Ье re­

presented висЬ that every independent route will Ьауе а step that belongs only to this route. ТЬеп the rate ofthis step w" is equal to that over the qth route, v(q)

(72)

Here Ь; and Ь; are the weights of the edges corresponding to the direct and reverse reactions of the 8th step, respectively, В; and Bi + 1 are the overaH weights for the spanning trees of the ith and (i + l)th graph nodes, respec­tively, accounting for the intermediates with the indices i and (i + 1) of the 8th step and В is the overaH spanning tree weight of the graph.

Recently, structurized forms for steady-state kinetic equations Ьауе Ьееп obtained that сап Ье written directly from the reaction graph [7-9]. Equation (69) is а particular саае for these structurized forms .

.---- --------._------------------------------

29

What does the application of the graph theory give? In the introduction to his book Zykov [10] эауэ: "Let иэ admit frankly: graph theory is not а revolution in science and the appearance ofthis apparatus in the mathemati­cal arena is not an encroachment оп mathematical fundamentals and does not introduce any ideas that равв the comprehension of most contemporaries nor wait for their complete recognition Ьу our descendants". His words сап Ье amplified Ьу the fact that, in physical chemistry, graphs have not proved revolutionary. Nevertheless it is а very useful method of representation. Reaction mechanism is аэ if it were created for geometrical representation. When drawing direct and inverse arrows in reaction equations, chemists did not know that in this way they draw а тар, а graph of complex chemical reaction. Concepts ofthe graph theory, namely "route", "circle", 'Ъаsis of circles" correspond in the most natural way to those of chemical kinetics. And it is not surprising that various authors are starting to иве graphs practicaHy in chemical and biochemical kinetics.

5. Elements of qualitative theory of differential equations

It is not our aim here to give а complete description of the fundamentals of а qualitative theory. This сап Ье found in the literature [11-18]. This section is only а short list of the basic concepts, i.e. а kind of а dictionary.

"Complex" behaviour of chemical reactions over а period of time сап Ье described only in terms of unsteady-state kinetic models. Ав а rule they are sets of ordinary differential equations of the type

~~ = [(е, k) (73)

Here с is the vector of variables and k the vector of parameters. U виаНу the right-hand side of eqn. (73) does not contain the time variable, t, in the explicit form. Such systems are called autonomous.

The phase эрасе of eqn. (73) is the врасе of vectors е. Its points are specified Ьу the coordinates C1 , ..• , Сп. The set of phase эрасе points is the set of аН possible states of the system. Phase эрасе сап Ье not only the whole vector врасе but also а certain part. Thus in chemical kinetic equations, variables are either concentrations or quantities of substances in the вуэ­tem. Their values cannot Ье negative. It is therefore natural to restrict ourselves to the set of those с аН the components of which are not negative, i.e. С; ~ о. In what follows we shall refer to these с values ав non-negative. Hence positive are those с values аН the components of which are positive, i.e. С; > о.

Chemical kinetic equations роввевв the following properties. For any non-negative initial conditions, со, the only solution of eqn. (73) that exists is C(t, k, ~). At the initial moment it takes values of со; i.e. С(О, k, со) = ~.

References рр. 45-46

30

Its solution is determined over the whole of the positive time half-axis t Е [О, (0), is а continuous and differentiable function of t, k, со and is non-negative at t )о О.

In chemical kinetics it is natural to consider the solutions of eqn. (73) for t )о О оп the positive half-axis [О, (0). Generally speaking, the function c(t, k, со) сап also Ье determined for negative values of t, but in this саве concentrations сап take negative values, i.e. they have по physical вепве. А рЬаве semi-trajectory (positive) will Ье called а curve in the phase врасе

whose points are the values of c(t, k, со) with fixed k, со and t Е [О, CXJ). If c(t, k, со) is also determined (and has а physical вепве) for t Е (- CXJ, О] опе must speak about negative semi-trajectories, i.e. values of c(t, k, со) for t Е (- 00,

О] and а whole trajectory, i.e. c(t, k, со) values for t Е ( - CXJ, (0). Movements of the point over the рЬаве trajectory or semi-trajectory account for the variations in the reactor mixture composition during the reaction. Since the equation с = c(t, k, со) Ьав only опе solution, every point ofthe рЬаве врасе is passed Ьу опе and only опе, of the рЬаве trajectories which neither intersect nor merge. А particular саве ofthe рЬаве trajectory is а rest point (а singular point), i.e. c(t, k, со) == со; 7(со , k) = О. If movement starts from the rest point, the system will remain there. For а closed chemical system, i.e. the system that does not exchange substance with the environment, the rest point is called ап equilibrium state. For ореп systems it is called а steady state. There exist specific trajectories, e.g. limit cycles, that are closed. ТЬеве specific points and trajectories have special fields of applica­tion. ТЬе totality ofphase trajectories is characterized Ьу the рЬаве portrait. It is the advantage ofthe qualitative analysis that it сап Ье used to represent а phase portrait for the set of equations without its solution since, in тапу cases, ап analytical solution cannot Ье obtained and the possibilities of computing methods are also limited. А special problem is to establish the number of rest points. It will Ье discussed in what follows primarily ав applied to heterogeneous catalytic reactions.

Important are the concepts of ап w-limit (ш in the вепве of the last опе, i.e. "from С( to ш") point and ап w-limit set. The point С1 is called the w-limit for the solution of eqn. (73) c(t, k, со) if the approach of this solution to С1 will Ье arbitrarily close after the arbitrarily high period of time. It теапв that there exists а sequence {t1 , ••• , tn, ... }, tn --> CXJ, such that сиn, k, со) --> c1 .

The set ofthe whole of w-limit points for си, k, со) is called ап w-limit set and is designated ав w(k, со), In the simplest саве, w(k, со) сап consist only of опе rest point [Fig. l(а)]. Тшв set itself сап Ье а closed trajectory, i.e. а limit cycle [Fig. l(Ь)]. А detailed analysis has Ьееп carried out for the systems with two variables (оп the plane) [11-14]. Hitherto three-dimensional systems had not Ьееп analyzed. In 1963, Lorenz [19] established that the seemingly simple set of three equations

С1 О'(Сl - С2 )

З1

(а)

о (Ь)

Fig. 1. Examples of w·limit sets. (а) Rest point; (ь) limit eyele; (е) Lorenz attraetor [projection оп the (е!, сз ) plane, о' = 10, r = ЗО, Ь = 8/З].

(:3 = - ЬСЗ + C1 С2

(where (J, r and Ь are parameters) has а limit set of unexpectedly complex character. The behaviour of this phase trajectory is illustrated in Fig. l(с). This limit set was called а Lorenz attractor. Limit sets similar to Lorenz attractors are called strange attractors. Ап important problem of the qualitative analysis is to elucidate the

structure of ш-limit sets. Unfortunately, there is по commonly accepted method to solve this problem. ТЬе exception is likely to Ье two савев studied extensively, i.e. two-dimensional and multi-dimensional systems but near the rest point. In the latter саве опе must first establish whether the rest point is stable and, in particular, whether it is ш-limitiпg for the initial data that are close to it. ТЬе concept of stability is of great importance for the qualitative theory, and Ьепсе it must Ье considered in more detail. There exists local stability, i.e. to sufficiently low perturbations, and global stabil­ity, i.e. to perturbations of а given value or (often) of ап arbitrary value. ТЬеве types of stability do not coincide. Having proved the local stability of the system, i.e. the existence of such 8 with which the system with perturba­tions lower than this 8 remains stable, we cannot yet вау that the system is stable in the global вепве. It сап Ье illustrated Ьу the simple mechanical analogy of а ЬаН at the bottom of а сир. If the walls of the сир are infinitely high, at апу deviation theball will roll down to the bottom (the asymptotic global stability). If the сир size is infinite, at втаll deviations the Ьаll will return to its initial position (the asymptotic local stability) but with а

References рр. 45-46

32

considerable displacement the Ьаll will slip out of the cup (i.e. there is по asymptotic global stability).

Let us give accurate definitions. Let c(t, R, со) Ье а solution for eqn. (73) which satisfies the initial condition сио) = со. This solution is called stable (according to Lyapunov) iffor апу infinitesimal G > О there exists values of д > О such that the inequality

Ico - Fol < д

results in

(74)

(75)

Here со and F о are systems variable at time t = О in the unperturbed and perturbed cases, respectively.

The solution C(t, k, со) is called asymptotically stable ifit is stable accord­ing to Lyapunov and there exists values of д > О such that the inequality (74) results in

at t ---+ 00 (76)

8ince rest points are particular cases of the phase trajectories c(t, k, со) == со, the аЬоуе definitions of stability according to Lyapunov are also valid for them. А rest point is stable according to Lyapunov if, for апу G > О there exists values of д > О such that after а deviation from this point within д, the system remains close to it (within the G value) for а long period oftime. А rest point is asymptotically stable ifit is stable according to Lyapunov and there exists values of д > О such that after the deviation from this point within д the system tends to approach it at t ---+ 00. Мапу other definitions for stability are known and they do not always

look alike. Each of them characterizes а required property of the solution under study. Though the Lyapunov definition of stability веетв to Ье the most natural, in тапу савев it cannot Ье used. No stability definition сап fit every real case. 8оте other versions of this concept are given in ref. 20 in which further references сап Ье found.

80 far we have defined the local stability ("there exists such д ав ... "). Now let us define the global stability for rest points. The rest point со is called globally asymptotically stable (ав а whole) within the phase врасе D ifit is stable according to Lyapunov, and for апу initial conditions (10 Е D the solution C(t, k, (10) tends to approach со at t ---> 00.

In what way is ап unsteady kinetic model investigated to elucidate wheth­er the rest point is locally stable? In this саве а combination of approaches, which сап Ье called а "rite", is used:

(1) То linearize the system: (а) to introduce new variables"t = с - с that are equal to the discrepancies between the current concentrations с and the concentrations с at а rest point; (Ь) in eqn. (73) to go from с to "t; and (с) to expand the right-hand side into а series over the powers of"t and to discard

33

the non-linear terms. The linearized system obtained is represented in the vector form Ьу

dt = p~ dt

where Р is the matrix of partial derivatives

Р = дf(с, k) I де

c=~

or in the coordinate form

d~i = I дЦс, k) l ~_ dt - дс- )

) ) ~ с =с

(2) То obtain а characteristic equation. It is of the form

det (Р - А 1) = о

where 1 is the unit matrix. (3) То investigate the roots of the characteristic equation.

(77)

(78)

(79)

(80)

The stability of the rest point for eqn. (73) depends оп the roots of the characteristic equation. The rest point is asymptotically stable if the real parts of аll the roots in eqn. (80) are negative. It is unstable if the real part of at least one of the roots is positive. In the саве where воте roots in eqn. (80) are purely imaginary and the rest of them have а negative real part, the stability cannot Ье established Ьу using only linear approximations. In this саве the rest point of eqn. (77) is stable but not asymptotic.

Let us illustrate the sense of the "rite" Ьу investigating the stability of а system with two variables. Let us take the system

(81) С 2 Q(c j , С2 )

After introducing the variables ~) = С) - (;1 and ~2 = С2 - (;2, the linearized system (81) is of the form

(82)

(83)

References рр. 45-46

34

А solution for eqn (82) is found in the form

A i.t е

After substituting eqn. (84) in eqn. (83) and reducing Ьу e)·t, we obtain

a l1 A + а12В

а21А + а22В

From eqn. (86) we сап write

А = _в а22 - Л

а21

and substituting this into eqn. (85) gives

[(a l1 - А)(а22 - А) - а12 а21 ] В = О

For В #- О we Ьауе

(84)

(85)

(86)

(87)

(88)

(89)

It is just the characteristic equation. According to eqn. (80) it сап Ье written ав

I all л а121 а21 а22 - А

о (90)

From eqn. (89) we obtain

л = a l1 + а22 + J(all + а22)2 1. 2 2 - 2 + а21 а12 - a ll а22 (91)

According to the theory of linear differential equations, the overall solution for eqn. (82) is found in the form

Cll е;'I' + С12 е ;'2'

С21 еЛ1 ' + С22 е)'2'

Here Си is determined Ьу the initial conditions.

(92)

ТЬе analysis of characteristic roots А determines not only the local stabil­ity (in the втаll) of а singular point for system (81), but also the character ofmotion near it, i.e. its type. Let us investigate the linear equation (82). For this purpose we will consider the following савев separately.

(1) The roots of the characteristic equation (89) аге real. Here several variants are possible. (а) А1 < О, А2 < О, and А1 #- л2 • ТЬе solution of eqn. (82) is а вuш of the

exponents ~1 = Clle )'lt + С12 е Л2t and ~2 = С21 е ~lt + С22 е ,(2

t, decreasing with

time. Consequently the zero solution (а rest point) i~ stable. Ттв singular

~-~--~~--~~~~~~~~~~~~~~~~~~-

35

(Ы (е)

(9) (h) (i)

Fig. 2. Турев of stationary points оп the plane. (а), (с), (е) Stable nodes; (Ь), (d), (f) unstable nodes; (g) saddle point; (h) stable focus; (i) unstable focus, (k) whirl.

point is called а stable node. ТЬе character oftrajectories near а stable node is illustrated in Fig. 2(а), where u is а straight 1ine specified Ьу the equation

~ = С11 ~2 С21

and v is а straight line specified Ьу the equation

~l Cl2 ~ = С22

In this case, to Ье specific we assume that).l < ).2' (Ь»).1 > О, A,z > О, and А,l "# А,2 (here, to Ье specific we assume that }'l > А,2)'

References рр. 45-46

36

Phase trajectories extend far from the singular point. It is an unstable node [Fig. 2(Ь)].

(с) А1 = А2 = А, А < О. Two different cases exist, either the matrix (83) is diagonal

[all a12] = [А О] а21 а22 О А

or it сап Ье reduced to the form

Ьу the linear transformation of variables. In the former case the solution is ofthe form ~l = Aei.t, ~2 = Bei

.t and the trajectories behave as shown in Fig.

2(3). In the latter case

The character of trajectories is illustrated in Fig. 2(c)where v is а straight line specified Ьу the equation ~1/~2 = С12/С22 ' In both cases the rest point is also called а stable node.

(d) А1 = А2 = А, А > О. This case differs from the previous one in reversing the direction ofmotion. Phase trajectories extend far from the singular point [Fig. 2(d), (f)] which is called an unstable node.

(е) The roots have opposite signs (e.g. А1 > О, А2 < О). The general solu­tion is the sum of exponents with opposite signs. The singular point is unstable, since with time the term with а positive exponent will Ье predomi­nant. This singular point is called а saddle [Fig. 2(g)]. There are only two trajectories moving over which it is possible to enter this point. Consequent­ly, in terms of more strict terminology, it should Ье treated as semi-stable.

(2) The roots have аn imaginary part. То analyze this case, eqns. (82) must Ье reduced to one second-order

equation. Differentiating the first equation and eliminating ~2 gives

d2~1 2 < d~l 2 !' О dt2 + U dt + ШО ~1 =

where

2д = (all + а22 ), ш0 2 = all a22 - а12 а21 Let us write eqn. (91) in the form

А1 • 2 = - (j ± (д2 - ш0 2 )У,

Since the roots have imaginary parts, шо 2 - (у2 = ш2 > О and

А1, 2 = - (j ± iш

(93)

(94)

(95)

(96)

Опе сап readily show that the solution for eqn. (93) is of the form

~(t) = e~bt(cjCOswt + c2sinwt)

Here the following савев are possible.

37

(а) д > О. The solution ~ (t) is in the form of damped oscillations. Phase trajectories are converging spirals and the singular point is а stable focus [Fig. 2(h)].

(Ь) д < О. Phase trajectories are diverging spirals. The singular point is ап unstable focus [Fig. 1.2(i)].

(с) д = О. The solution is undamped oscillations with frequency ш. Phase trajectories are enclosed еШрвев [Fig. 2(k)]. The singular point is called а whirl. It is the non-rough point, i.e. even with insignificant variations in the parameters, the phase picture alters. For the rough point, the phase picture is not affected Ьу втаН variations of parameters. The above terms (rough and non-rough) were introduced Ьу Andronov et al. [11, 12].

Closed trajectories around the whirl-type non-rough points cannot Ье mathematical models for sustained self-osci1lations since there exists а wide range over which neither amplitude nor self-oscillation period depends оп both initial conditions and system parameters. According to Andronov et al., the stable limit cycles are а mathematical model for self-oscillations. These are isolated closed-phase trajectories with inner and outer sides approached Ьу spiral-shape phase trajectories. The literature lacks general approaches to finding limit cycles.

There exists the concept of "Bendikson criterion". If, for eqn. (81), the expression

aP(c j , с2 ) aQ(c j , С2 ) --~~~ + ~~~~

дс! дС2

does not change its sign in а certain region of the phase plane, this region contains по closed-phase trajectories.

In principle, to study the local stability of а stationary point from а linear approximation is not difficult. Some difficulties are met only in those савев where the real parts of characteristic roots are equal to zero. More соm­plicated is the study ofits global stability (in the large) either in а particular preset region or throughout the whole phase врасе. In most савев the global stability сап Ье proved Ьу using the properly selected Lyapunov function (а so-called second Lyapunov method). Let us consider the function V(c) having first-order partial derivatives д V/ дс;. The expression

(97)

will Ье referred to а derivative V of eqn. (73) and denoted Ьу V. This expression сап Ье treated ав а derivative V of the solution of eqn. (73)

References рр. 45-46

38

(98)

The second Lyapunov method implies that one ивев V values which have maxima at the rest point under study whose derivative [eqn. (98)] is not positive CV :( О) in the vicinity of this point and zero values are admitted only at this point.

From the various versions of this method we will choose only опе. Let V :( О and, only at the rest point under study с, V = о. Then let V have its minimum, V(e) = Vmin at the point с and for воте 8 > Vmin the set specified Ьу the inequality V~o) < 8 is finite. Then for any initial conditions со from this set the solution of eqn. (73) is c(t, k, со) -> с at t -> со. V~) is called а Lyapunov function. The arbitrary function whose derivative is negative Ьесаиве of the system is called а Chetaev or sometimes а dissipative func­tion. Physical examples are free energy, negative entropy, mechanical en­ergy in systems with friction, etc. Studies of the dissipative functions сап often provide useful information about а given system. А modern representa­tion for the second Lyapunov method, including а method of Lyapunov vector functions, сап Ье found in ref. 20.

In addition to Lyapunov functions, it is also useful to investigate co-in­variant sets. The set В in the phase врасе is called co-invariant provided that, for any solution of eqn. (73), сщ due to сио) lies within В, i.e. сио) Е В, it follows that for any t1 > to , c(t1 ) ЕВ. The co-invariant set is а type of 'CЬag": once entered there at воте moment of time, the solution will not leave at any subsequent moment. It is evident that, for any dissipative function, the V-set specified Ьу the inequality Щс) < 8 is co-invariant. Let V~(to» < 8, but due to dissipativity for any t1 > to V(c(t1» :( V(c(to» and certainly V~(t1» < 8.

In what follows, however, we will meet co-invariant sets that are not associat­ed with апу differentiated dissipative function. The simplest example of an co-invariant set is а positive semi-trajectory.

Along with аН its points, апу co-invariant set has а positive semi-trajec­tory coming from it. Therefore among the co-invariant sets containing а given point а positive semi-trajectory coming from it will Ье minimal. We believe it will Ье of interest for systems, depending оп parameters, to inves­tigate co-invariant sets that are independent of them (or of а certain part of the parameters). Here the minimum co-invariant sets containing the point со wiH по longer Ье semi-trajectories. These sets [designated here ав щсо )] consist of those points с for which there exist such moments of time t1 > О, and the function k(t) оп the section [О, t1 ], taking values within the region of parameters for the initial eqn. (73) such that the solution сщ for the equation

1~, k(t» (99) dt

39

with the initia1 conditions с(О) = со, at the initia1 moment oftime takes the va1ue с: C(t1 ) = с. In other words, among the ш-invаriаnt sets that are independent of а certain series of parameters and contain а given point, the minimum set consists of those с into which the solution of type (99) equa­tions with the initial conditions с(О) = со сап fall at а positive moment of time. In this case the parameters ofwhich V(co) must Ье independent сап Ье arbitrary time functions. In constructin!5 V(co) we deal with so-called dif­ferential inc1usions: the derivative dc/ dt in eqn. (99) has not been specified. We preset on1y the set to which it be1ongs. А theory of differentia1 inc1usions has been deve10ping largely for the app1ication to problems of automatic monitoring and contro1 when the information about some object is incom­p1ete. We will not need the data accumu1ated in this fie1d since differentia1 inc1usions in kinetics are rather specific.

An important property of the systems having а convex finite ш-invаriаnt set is as follows. Any c10sed convex finite ш-invаriаnt set has а steady-state point. This follows from the known Brower fixed-point theorem (see, for examp1e, ref. 21), that has been extensively used in various fields of mathe­matics to prove theorems concerning the existence of solutions.

In conc1usion of our short excursion into the qualitative theory of dif­ferentia1 equations, we shall discussed the often-used term 'Ъifurсаtiоn". It is ascribed to the systems depending оп some parameter and is app1ied to point to а fundamenta1 reconstruction of phase portrait when а given par­ameter attains its critical value. The simplest examp1es of bifurcation are the appearance of а new singular point in the phase plane, its 10ss of stability, the appearance (birth) of а limit cycle, etc. Typical cases оп the p1ane have been discussed in detail in refs. [11, 12, and 14]. For higher dimensions, по such studies have been carried out (and we doubt the pos­sibility of this).

Wherever possible, а qualitative analysis of equations must precede num­erical simulation. However, particularly for dimensions greater than 2, it is not always possible to carry out а complete analysis without а computer. Therefore а tendency has recently developed to apply numerical calcula­tions to "guess" qualitative peculiarities. А reasonable strategy for сот­plicated cases is to combine numerical experiments and their "qualitative" interpretation.

6. Relaxation in catalytic reactions

Let us consider the catalytic isomerization reaction whose steady-state kinetic model has already been considered in the previous section. Its de­tailed mechanism is of the form: (1) А + Z +± AZ; (2) AZ +± BZ; (3) BZ +± В + Z. Under the assumption of constant concentrations of substan­ces in the gas phase, it will Ье written as

References рр. 45-46

40

d~~] = _ kj[A][Z] + k_j[AZ] + kз[ВZ] - k_З[В][Z]

d[AZ] dt

(100)

where the concentrations [А] and [В] are the parameters. А solution is found to Ье [Z]" = ае-;", [AZ] = Ье- 1', [BZ] = се-Л'. We

obtain

- kj[A]ae- i., + kjbe-i-t - kзсе-;" - k_з[В]ае- lt

k j[A]a8''' - k_ 1be- lt - kzbe-i-' + k_zce- M

kzbe-i.' - k_zce-i.' - kзсе- i., + k_з[В]ае- i.,

Dividing eqns. (101) Ьу е-а

a(kj[A] + k_з[В] - А) - bk_ j - сk з О

о

- аk_з[В] - bkz + с(kз + k_ z - А) О

(101)

(102)

The set of equations (102) сап Ье represented as а matrix. Elements of the whole of its columns have co-factors а, Ь and с, respectively. Dividing the columns Ьу these

kj[A] + k_З[В] - А - k_ j - kЗ

= о (103)

It is just the characteristic eqn. (80) for а given case. Let ив write the determinant of matrix (103) ав а sum of the products for the elements of the first row with their algebraic complement

(kj[A] + k_З[В] - А) . + [

kz + k_ j - А - k_ 2 ]

- k z kЗ + k_ z - А

+k' j 2 -k. 1 =0

1- k [А] - k 1 1- k [А] k 2 + k_ j - АI

-1 _ k._З[В] kЗ + k_2

- А 3 - k_з[В] - k2

(104)

whence it follows

АЗ - AZ(b j + bz +Ьз + Ь_ ! + b_ z + Ь_ З ) + (Bz + BAZ

+ BBZ) = о

(105)

41

We obtain three characteristic roots

Ь) + Ь2 + Ьз + Ь _) + Ь -2 + Ь_ З (106)

А)А2 = B Z + B AZ + BBZ

It is interesting that the product of the characteristic roots is the БUПl of the whole ofthe trees in the graph for this reaction mechanism [Бее eqns. (60) and (64)].

Since опе of the roots is zero, we obtain

[Z] и)

[AZ] (t)

[BZ] (t)

[AZ] + /3)еЛ1t + /32еЛ2t

[BZ] + У) еЛ1t + уzеЛ2t

(107)

Here [Z], [AZ] and [BZ] are the steady-state concentrations ofthe respective substances: [Z] = [Z](t), [AZ] = [AZ](t), [BZ] = [BZ](t) at t -> 00.

Опе must remember опе important thing. The values of А determining' relaxation are not rate constants аБ such. In general, the characteristic roots А are rather complex functions of these constants. Апу dynamic system ЬесотеБ stable eventually and сотев to the rest

point, i.e. attains its equilibrium or steady state. For closed systems, а detailed equilibrium is achieved at this point. This is not во simple аБ it would веет, аБ substantiated Ьу а principle of the thermodynamics of ir­reversible processes. At а point of detailed equilibrium not on'ly does the substance concentration remain unchanged (dc/dt = О), but also the rate of each direct reaction is balanced Ьу that of its associated reverse counterpart (ш +; = W _).

А necessity to apply this principle will Ье illustrated Ьу а simple example. Let иБ take а triangle of the reactions

/З:А~ С ... 2 В

-2 ~ (108)

Note that the reaction С <=± А is а combination of А <=± В and В <=± С. Mechanism (108) corresponds to the kinetic model

d[A] dt d[B] dt = - (k_ 1 + kz)[B] + k_ 2 [C] + k1 [A]

References рр. 45-46

42

(109)

At equilibrium аН the derivatives are equal to zero. From eqns. (109) опе сап readily obtain

[1\] (k_JB] + kз(С])

(110) k1 + k_ з

[В] (k_z[C] + k1[1\])

(111) k_ 1 + kz

[С] (kz[B] + k_ з [1\]) (112)

k_ z + kз

Here [1\], [в] and [С] are the equilibrium concentrations of the respective substances. After eliminating [С] from eqns. (110) and (111), it сап easily Ье determined that

[В] [1\] =

Ву analogy

[С] [В]

[1\] [С]

k1k_ z + k_зk_ z + k1kз kzkз + k_1k_ з + k_1k_ z

k1kz + k_зk_ 1 + k_зkz k-zk1 + k_zk_ з + k1kз

kzkз + k_1kз + k_1k- z k1kz + k_зk_ 1 + k_зkz

(113)

(114)

The form of eqns. (113) and (114) is rather strange, since ratios of equi­librium concentrations must equal equilibrium constants.

[В] k1 = КР1 [1\] k_ 1

[С] kz = K pZ [В] k_ z

[1\] kз = КрЗ [С] k_ з

After multiplying these relationships, we obtain а new equation

k1 k2 kз КРIКРZКРЗ =k k k = 1

-1 -z -3

(115)

(116)

This equation imposes limitations оп kinetic model parameters. Whence

k 3 = k1kzkз - k_

1k_

2 (117)

After substituting eqn. (117) into eqn. (113), we obtain

[в] k 1k_ 2 + k1kз + k_2(klk2kз/k_lk_2) [1\] kз k2 + kзk_ j + k_ 1 k_ 2

k j (k_ 2k_ j + kзk_ 1 + k2kз ) k_ 1 (kз k2 + kз k_ 1 + k_ 1k_ 2)

k1 -k = КР1

-1

43

(118)

Thus to fit ordinary equilibrium relationships, system parameters should Ье related Ьу an equation оНЬе eqn. (117) type. It was Wegscheider in 1902 who first noted this fact in Ыв study of а reaction that is more complex than the аЬоуе, i.e. (1) А1 <:z А2; (2) Аз <:zA4; (3) А2 + Аз <:z А} + А4•

Detailed equilibrium must occur in closed systems, whereas in open systems, particularly in those that are far from being in equilibrium due to their exchange with the environment, the situation is much more complicat­ed. Primarily, steady-state solutions сап Ье multiple, i.e. the rates of sub­stance formation and consumption сап Ье balanced оп many points.

For several савев, e.g. for linear pseudo-steady-state equations (linear mechanisms), the steady state is certain to Ье unique. But for non-linear mechanisms and kinetic models (which are quite common in catalysis, e.g. in the саве of dissociative adsorption), there тау Ье several solutions. Multiplicity of steady-states is associated with types of reaction mechan-1втв.

Fairly recently it has been established that а set of pseudo-steady-state equations for complex catalytic reactions сап Ьауе several solutions only when their detailed mechanisms involve ав one step an interaction between various intermediates [22]. ТЬе simplest catalytic mechanism possessing this property is an adsorption mechanism. For example

02 + 2Pt ..... 2PtO

СО + Pt ..... PtCO

PtO + PtCO ..... 2Pt + С02

Let us write pseudo-steady-state equations for mechanism (119)

2k1 [02](1- [PtO] - [PtCO]i - kз[РtСО][РtO]

k2[CO](1- [PtO] - [PtCO]) - kз[РtСО][РtO]

о

о

(119)

(120а)

(120Ь)

ТЬе value 1 - [PtO] - [PtCO] is equal to the concentration of unoccupied sites оп the Pt surface.

ТЬе system (120) with certain value во the parameters сап have several steady-state solutions, i.e. so-called boundary solutions

[PtO]

[PtO]

1, [PtCO]

о, [PtCO]

References рр. 45--46

о

1

44

In the former саве the whole оНЬе surface is covered Ьу oxygen, in the latter саве it is covered Ьу СО. Here the reaction rate is zero. Other solutions also exist. After subtracting eqn. (120Ь) from eqn. (120а), we obtain

2k j [02](1 - [PtO] - [PtCO])2 = k2 [СО](l - [PtO] - [PtCO]) (121)

whence

[Pt] = 1 - [PtO] [PtCO] (122)

The reaction rate is

(123)

There are two steady states with this reaction rate since eqns. (120а) and (120Ь) are symmetrical with respect to [PtO] and [PtCO]. ТЬе analysis indica­tes that the steady states with non-zero rates are observed only when the conditions

2k j [02] > k2 [CO]

kg(2k j [02] - k2[CO])2 > 8k j [02] k22[CO]2

(124)

are met. Thus there exist four steady-state solutions two of which are stable and

the other two unstable.

kHCO]2 w= --2k j [02]

provided that the conditions (124) are met and the initial state of the surface is within the attraction region of the internal stable study state and

w = о

for the remaining савев. With fixed [СО], and increasing [02] the steady-state reaction rate W is

initially zero (the overall surface is covered with СО) and it сап then jump to the value k 2 2[СО]2 /2k j [02]' With further increase in [02]' the reaction rate varies inversely with [02]' In turn, with constant [02] the reaction rate rises quadratically with increasing [СО] and then "jumps" down to zero values. ТЫв example indicates that rather simple but non-linear schemes сап Ье characterized bycomplex dynamic behaviour. In radiophysical terms, scheme (119) сап Ье called а simple catalytic trigger since, in this саве, there exist two stable steady states.

The kinetic model accounting for the adsorption mechanism (119) cannot have rate self-oscillations since, in this саве, Bendikson criterion is met

д (d[PtOJ) д (d[PtCO]) a[ptO] dt + a[PtCO] dt < О (125)

45

If one wants to describe the observed experimental rate self-oscillations, mechanism (119) must Ье completed Ьу а step with at least one more inter­mediate or the rate constants for the steps must Ье assumed to Ье dependent оп the concentrations [22, 23].

ТЬе аЬоуе kinetic models are based оп the surface action law that is absolutely analogous to the тавв action law for volume reactions in ideal systems. In this саве а model of"an ideal adsorbed layer" acts, which is valid under the following assumptions:

(1) the catalyst is homogeneous, i:e. аН its components are energeticaHy the вате, and the chemisorption energy is independent of the surface coverage with various substances;

(2) the catalyst is stable and its properties do not depend оп the composi­tion of the reaction and its effect оп the catalyst; and

(3) the energy distribution in the system is equilibrium. А large number of experiments (calorimetric, isotopic, etc.), however,

indicate the considerable energy inhomogeneity of catalysts. Оп the other hand, the literature reports numerous experimental data testifying to the fact that catalysts themselves significantly change under the effect of reac­tion media. Ттв was especiaHy emphasized Ьу Boreskov [24].

Boudart [25] noted that "though the non-ideal nature of аН real catalytic surface Ьав been convincingly proved, it remains unclear what form their associated equations must take". Boreskov [24] states: "One сап note а striking contradiction between multiple савев of inhomogeneous real cat­alysts and the втаН portion of kinetic equations accounting for their inho­mogeneity". Apparently, the "ideal adsorption layer" model wherein the catalyst surface is treated ав а chessboard, in several савев proves to Ье too simplified. Great efforts are to Ье made to construct physicaHy substantiated models that will account for the complex totality of аН the processes taking place in the "reaction mixture-catalyst" systems.

References

V.I. Dimitrov, Simple Кinetics, Nauka, Novosibirsk, 1982 (in Russian). 2 J. Horiuti and Т. Nakamura, Z. Phys. СЬеm. N.F., 11 (1957) 358. 3 M.I. Temkin, Dokl. Akad. Nauk SSSR, 152 (1963) 156. 4 M.I. Temkin, Zh. Vses. ктm. Оуа., 20 (1975) 7. 5 Е. King and С. Altman, J. Phys. СЬеm., 60 (1956) 1375. 6 M.V. Volkenshtein and B.N. Goldshtein, Biokhimiya, 31 (1966) 541; Dokl. Akad. Nauk

SSSR, 170 (1966) 913. 7 V.A. Evstigneev and G.S. YabIonskii, Кinet. Katal., 20 (1979) 1549. 8 G.S. YabIonskii and V.I. Bykov, Kinet. Katal., 18 (1977) 1561. 9 V.A. Evstigneev, G.S. YabIonskii and V.1. Bykov, Dokl. Akad. Nauk SSSR, 245 (1979) 871.

10 А.А. Zykov, Finite Graphs Theory, Nauka, Novosibirsk, 1969 (in Russian). 11 А.А. Andronov, Е.А. Leontovich, 1.I. Gordon and A.G. Maier, Qualitative Theory of

Second-Order Dynamic System, Nauka, Moscow, 1966 (in Russi~n).

46

12 А.А. Andronov, Е.А Leontovich, 1.1. Gordon and AG. Maier, Theory of Bifurcation of Dynamic Systems оп а Plane, Wiley, New York, 1973.

13 А.А. Andronov, АА Vitt and S.E. Khaikin, Theory of Oscillations, Fitzmatgiz, Moscow, 1959 (in Russian).

14 V.I. ArnoId, Additional Chapters ofthe Theory ofOrdinary Differential Equations, Nauka, Moscow, 1978 (in Russian).

15 N. Minorsky, Nonlinear Oscillations, Van Nostrand, Princeton, NY, 1962. 16 N.N. Bogoljubov and U.A Mitropolskii, Asymptotic Methods in the Theory of Non-linear

Oscillations, Nauka, Moscow, 1974 (in Russian). 17 B.D. Hassard, N.D. Kazarinoff and J.-H. Wan, Theory and Applications of Hopf Bifurca­

tion, Cambridge University Press, London, 1981. 18 М. Kubicek and М. Marek, Computational Methods in Bifurcation Theory and Dissipative

Structures, Springer-Verlag, New York, 1983. 19 E.N. Lorenz, J. Atmos. Sci., 20 (1963) 130. 20 А.А. Martynjuk and R. Gutovskii, Integral Inequalities and Stability ofMotion, Naukova

Dumka, Kiev, 1979 (in Russian). 21 М.А. Krasnose!skii and Р.Р. Zabrejko, Geometrica! Methods of Non!inear Ana!ysis,

Nauka, Moscow, 1975 (in Russian). 22 M.G. Slin'ko, V.I. Bykov, G.S. Yablonskii and Т.А. Akramov, Dokl. Akad. Nauk SSSR, 226

(1976) 876. 23 M.G. Slin'ko and М.М. Slin'ko, Catal. Rev. Sci. Eng., 17 (1978) 119. 24 G.K Boreskov, Kinet. Katal., 13 (1972) 543. 25 М. Boudart, Usp. Khim., 43 (1974) 317 (in Russian).